g p s f o r mat 0 6 5 0 a workbook of problems for college ...xii chapter 12: verbal problems...
TRANSCRIPT
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G P S F O R MAT 0 6 5 0
A Workbook of Problems for College Algebra
By
Prof. J. Greenstein
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TABLE OF CONTENTS
“‘Begin at the beginning’, the king said, very gravely, ‘and go till you come to the end:
then stop’”.—Lewis Carroll
Preface
I Chapter 1: Introduction
Section 1.1: Nomenclature-Terms, Constants, Monomials, Binomials,
Trinomials, Absolute Values
Section 1.2 Exponents- aN × aM; (ab)N; (a/b)N; aN/aM; a0 ; a-N
Section 1.3 Scientific Notation
II Chapter 2: Signed Numbers
Section 2.1 Operations involving Multiplication; Division; Addition;
Subtraction
Section 2.2 Order of Operations
III Chapter 3: Evaluation
Section 3.1 Evaluating Algebraic Expression
Section 3.2 Functional Notation
IV Chapter 4: Multiplication of Algebraic Expressions
“I wish I didn’t know now, what I didn’t know then.”
Section 4.1 Multiply Monomial X Monomial X Monomial
Section 4.2 Monomial X Binomial—the Distributive Law
Section 4.3 Monomial X Trinomial
Section 4.4 Binomial X Binomial; (Binomial)2
Section 4.5 Binomial X Trinomial; (Binomial)3
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V Chapter 5: Combining Like Terms
VI Chapter 6: Factoring
Section 6.1 Unique Prime Factorization of Integers
Section 6.2 Greatest Common Monomial Factor (GCF)—Numerical; Variable
Section 6.3 Difference of Two Squares (DOTS)
Section 6.4 Combination of Greatest Common Monomial Factor and Difference
of Two Squares
Section 6.5 Trinomial with Leading Coefficient 1 (a=1): ax2 + bx + c
Section 6.6 Trinomial with Leading Coefficient Other than 1 (a≠ 1):
ax2 + bx +c
Section 6.7 Combination of Greatest Common Monomial Factor and
Trinomial
Section 6.8 Factoring by Grouping
Section 6.9 Review of All Factoring
VII Chapter 7: Solving Equations and Inequalities
Section 7.1 Introduction to Equations- Is a Given Number a Solution
Section 7.2 Linear (First Degree) Equations
Section 7.3 Decimal Equations
Section 7.4 Fractional Equations—Monomial in the Numerator; Binomial in
the Numerator
Section 7.5 Literal Equations
Section 7.6 Quadratic (Second Degree) Equations
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Section 7.7 Inequalities—Showing Solutions algebraically and on the number
line
VIII Chapter 8: Fractions
Section 8.1 Simplifying Fractions—Monomial/Monomial;
Polynomial/Monomial; Polynomial/Polynomial
Section 8.2 Multiplying Fractions
Section 8.3 Dividing Fractions
Section 8.4 Combining Fractions—Like Denominators; Different
Denominators
IX Chapter 9: Roots and Radicals
Section 9.1 Pythagorean Theorem
Section 9.2 Simplifying Radicals—Numerical; Variable
Section 9.3 Multiplying Radicals—Monomial × Monomial; Monomial × Binomial
Section 9.4 Dividing Radicals- √𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛; Radical Divided by a Radical
Section 9.5 Combining Radicals- Like Radicands; Unlike Radicands
X Chapter 10: Graphing Linear Functions
Section 10.1 Introduction- The Rectangular Coordinate System; Plotting
Points
Section 10.2 Table Method
Section 10.3 Intercepts Method
Section 10.4 Slope – Y Intercept Method
XI Chapter 11: Solving a System of Two Equations In Two Unknowns
Section 11.1 Solving Graphically
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Section 11.2 Solving by the Substitution Method
Section 11.3 2 Solving by the Elimination Method
XII Chapter 12: Verbal Problems
Section 12.1 Ratio and Proportion Problems
Section 12.2 Percent Problems
Section 12.3 Number Problems
Section 12.4 Distance = Rate × Time Problems
Answers to the Odd Numbered Problems
“Where of one cannot speak, there of one must be silent.”
PREFACE
Many years ago, a man in Australia made a bet with a friend that he could eat
an entire Dodge Dart automobile in less than one year’s time –this was excluding the
rubber and glass parts, of course. When asked how he thought he could do it
without getting sick and dying, the man replied that he would cut the car into very,
very small pieces and eat it that way. The man was able to accomplish this task and
won his wager.
In this textbook, we will follow the “Dodge Dart” approach. The mathematical material will be presented in very, very small pieces. It is hoped that in this way, you
will be able to easily digest the mathematical material presented.
It is our objective that you obtain a great understanding of the elementary
algebra presented herein. If you don’t obtain a great understanding, it is hoped that
you receive a small understanding of the material. If you don’t receive a small understanding, it is hoped that at least it doesn’t make you sick.
Enjoy
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Section 1.1: Nomenclature--Terms, Constants, Monomials,
Binomials, Trinomials, and Absolute Values
“Algebra is the use of symbols to represent math things: numbers, lines, unknowns; and it is the study of rules for combining/working with these
symbols.”
Identify the coefficients and variables in each of the following: 1) 8𝑥2 2) −2𝑦3
3) 7𝑥2𝑦 4) 𝑦𝑧 5) 12𝑥𝑦𝑧 6) 2𝑥2 + 4
Classify each of the following as monomials, binomials, or trinomials: 7) 2𝑥 8) 6𝑥2 − 9 9) 3𝑥2 10) 𝑥8 + 45 11) 4𝑥2 + 7𝑥 − 11 12) 𝑥4 − 3𝑥2 + 𝑥
Identify each term in the following algebraic expressions: 13) 4𝑥2 + 8𝑥 − 10 14) 2𝑥4 − 9 15) −6𝑥3 + 7𝑥2 − 3𝑥 + 4 16) 19𝑥4 − 12𝑥3 + 10𝑥2 − 5𝑥 + 1
Identify the variable terms and the constant terms in the following 17) 10𝑥2 − 7𝑥 + 2 18) 6𝑥3 + 10 19) −7𝑥4 + 3𝑥2 − 4𝑥 + 17 20) 9 − 10𝑥 + 4𝑥2
Identify the degree, number of terms, and constant terms for: 21) 3𝑥2 + 8𝑥 − 7 22) 4𝑥5 + 2𝑥 + 3 23) 9𝑥16 − 4𝑥12 − 5𝑥 − 21 24) 4𝑥8
24) 2𝑥7 − 4𝑥6 + 9𝑥5 + 6𝑥3 − 7𝑥2 26) 2𝑥 − 24
Evaluate each of the following 27) |−4| 28) |9| 29) |−24| 30) |41|
31) |3| 32) |
−4|
8 5 33) −|11| 34) −|−12|
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Section 1.2: Exponents
“The only way to learn mathematics is to do mathematics. The problem is not
the problem. The problem is your attitude about the problem.”
Simplify:
1) 32 × 34
3) (−4)2
5) −42
7) 𝑥2 ∙ 𝑥6
9) 𝑚5 × 𝑚6
11) (𝑎7𝑏9)(𝑎3𝑏6)
13) (5𝑥2𝑦3)(−4𝑥5𝑦2)
15) −7𝑑𝑓2(−3𝑑2𝑓3)
17) 2(𝑥2)3
19) (3𝑧4)4
21) (2𝑎2𝑏3)4
𝑎6
23) 𝑎4
𝑐5
25) 𝑐
𝑝3
27) 𝑝9
2) 23 × 24
4) (−4)3
6) −43
8) 𝑦4 × 𝑦8
10) (𝑥3𝑦4)(𝑥2𝑦6)
12) (𝑟2𝑠3𝑡) × (𝑟3𝑠4𝑡2)
14) −6𝑝𝑞2(3𝑝3𝑞)
16) (2𝑥2)3
18) (5𝑦3)2
20) (𝑚5)6
22) (5𝑎4𝑏6)3
𝑏7
24) 𝑏2
𝑏4
26) 𝑏7
𝑞7
28) 𝑞12
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29) 𝑟−2
𝑡−3
31) 𝑡5
(𝑤5)3
33) 𝑤2
𝑦4
35) (𝑦3)5
37) (3⁄4)−2
39) (5⁄6)−2
41) (−4)−2
43) −(−4)−2
45) 𝑏−3 × 𝑏4
47) (4𝑟−3)2(2𝑟−4)3
49) (−5𝑥2)4
51) (5𝑥2𝑦−4)3
53) 2𝑎2(4𝑎2)(−2𝑎)
𝑠−3
30) 𝑠4
32) 𝑣5
𝑣5
(𝑥2)6
34) 𝑥5
𝑧−5
36) (𝑧−3)4
38) (2⁄3)2
40) 4−2
42) −4−2
44) 𝑎−4 × 𝑎−6
46) (𝑎3𝑏−5)3
48) (2𝑝2𝑞−4)−3
50) −5(𝑥2)4
52) 6𝑥3(4𝑥2)(−2)
54) 7𝑏(3𝑏2)(−1)
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Section 1.3: Scientific Notation
“Look not at the flask, but at what it contains.”
Convert into Scientific Notation 1) 2,400 3) 280 5) .003 7) .00000045
Convert into standard form 9) 4.8 × 105
11) 8.14 × 109
13) 9.4 × 10−2
15) 3.56 × 10−1
2) 3,600,000 4) 410,000 6) .05 8) .000874
10) 7.43 × 103
12) 3.14 × 102
14) 1.44 × 10−5
16) 6.18 × 10−8
Multiply and leave answer in standard form 17) (2 × 104)(3 × 103) 18) (6 × 107)(4 × 105) 19) (8.1 × 1012)(5.2 × 104) 20) (4.3 × 103)(3.6 × 108) 21) (4 × 10−2)(2 × 10−6) 22) (7 × 104)(3 × 10−10) 23) (6.9 × 10−2)(3.4 × 10−6) 24) (4 × 103)(4 × 10−3)
Divide and leave answer in standard form (6×104)
25) 26) (2×102)
(9×10−6) 27) 28)
(6×10−3)
(4.6×103) 29) 30)
(2.3×10−4)
(8.4×106)(5.1×104) 31) 32)
(4.2×105)
(1.4×109) 33) 34)
(2.8×104)(5×1012)
(8×105)
(5×10−6)
(4×10−7)
(5×102)
(3.4×104)
(1.7×109)
6.4×1015)(9×104)
(4.5×108)
(4.8×10−8)
(6.1×10−2)(2.4×10−11)
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Section 2.1: Operations Involving Addition, Subtraction,
Multiplication or Division
“There are three types of students: those who can count and those who can’t.”
Multiply:
1) 5(6) 2) 4 × 3 3) −7(3) 4) −6 × 4 5) (−2)(5) 6) 3(−10) 7) 5 × (−9) 8) (7)(−8) 9) (−7)(−3) 10) −8(−4) 11) −11 × (−3) 12) 4(2)(5) 13) 3(−2)(6) 14) 7(−3)(−2) 15) −8(−4)(−5) 16) −3(−2)(−8)(−1)
Divide:
36 17)
6
48 18)
12
−24 19)
3
−16 20)
8
22 21)
−11
50 22)
−5
−12 23)
−6
−18 24)
−9
25) 63 ÷ 7 26) −30 ÷ 15
27) 28 ÷ (−14) 28) (−100) ÷ (−10)
0 29)
9
7 30)
0
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Combine (Add or Subtract):
31) 4 + 6 32) 8 + 7 33) −9 + 14 34) −11 + 6 35) 12 + (−5) 36) 19 + (−8) 37) (−10) + (−11) 38) (−17) + (−7) 39) 17 − 5 40) 14 − 6 41) 12 − (−8) 42) 9 − (−6) 43) −3 − 4 44) −2 − 11 45) −10 − (−5) 46) −8 − (−7) 47) 4 + 8 + 6 48) 14 + 9 + (−5) 49) 10 + (−6) + (−9) 50) −12 + (−7) + (−3) 51) −15 + 8 + 7 52) −19 + (−10) + 8 53) −9 + 12 + (−3) 54) 2 + 6 − 4 55) 7 + 10 − (−5) 56) 18 − 7 + 6 57) 22 − (−3) − (−7) 58) −13 − 9 − (−11) 59) 4 + (−8) − (−7) + (−6) 60) 10 − (−2) − (4) + (−10) 61) −12 + (10) − (−9) − (3) 62) −15 − (−7) − (−9) − (−5) + 9
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Section 2.2: Order of Operations
“Mathematics is not a spectator sport. You have to do it to appreciate it, and doing it
Evaluate:
1) 4 + 5(6)
3) 8 + 4(−2)
5) 7 − 4 ÷ 2
7) (2 + 3)4
9) 5 ∙ 23
11) 8 − 16 ÷ 23
13) 35 ÷ 5 + 4 × 3
15) 22 − 5 × 2 + 24
17) 10 − 6(5 − 2)⁄9 (42−9)
19) (−7+4(0)−69−3)
(−4−8) 21)
(−5−1)
(−3(−5)−(−6)(−4) 23)
−3(8−5)
(−5−(−3)2) 25)
−7
requires patience and persistence.”
2) −2 + 3(5)
4) −8 − 5(−2)
6) 15 − 10 ÷ 5
8) (10 − 8)6
10) −4 ∙ 32
12) 18 − 32 ÷ 42
14) 20 ÷ 2 − 32 − 4 ∙ 2
16) 24 + 14 ÷ 7 − 8
18) 3(−4)2 + 25⁄(8 − 3)2
(25+6) 20)
(−2+4(2)−6)
(−14+(−6)) 22)
2(−5)
(4(−7)+2(7−3)) 24)
4(−5)
26) (12+3)
− (2−6)2
(4+1) 4
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(32−(7−8)3) 42−6 27) 28)
5(−2)+4(6−3) −5−3(8−2)
(−2)3−3(−4) −2(5−6)2
29) 30) 2(−2) −6+4∙3
4−32 10−22
31) −8(5 − 3) + 32) 10(9 − 4) + 5 2(−3)
33) 5(2 − 6)2 + 7 × 0 − (4 − 5)2 34) 11 − 4(3 + 2) − 6 ∙ 0 − (−2)3
35) 20 − 24 ÷ 8 + 6 36) 6 − 15 ÷ 3 × 5
37) 35 ÷ 7 × 3 − 5 38) 40 ÷ 23 × 3 − 9
39) 8(−7 − 5) 40) 8 − 7(−5)
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Section 3.1: Evaluating Algebraic Expressions
“Algebra symbols are used when you don’t know what you are talking about.”
Evaluate:
1) 4𝑥 𝑤ℎ𝑒𝑛 𝑥 = 3
2) −2𝑥 𝑤ℎ𝑒𝑛 𝑥 = 5
3) −6𝑥 𝑤ℎ𝑒𝑛 𝑥 = −3
4) 7𝑥2 𝑤ℎ𝑒𝑛 𝑥 = 2
5) 3𝑥2 𝑤ℎ𝑒𝑛 𝑥 = −4
6) 2𝑥2𝑦 𝑤ℎ𝑒𝑛 𝑥 = 4, 𝑦 = −2
7) 4𝑥2𝑦2 𝑤ℎ𝑒𝑛 𝑥 = −3, 𝑦 = −1
8) 2𝑥3 𝑤ℎ𝑒𝑛 𝑥 = −2
9) 𝑥4 𝑤ℎ𝑒𝑛 𝑥 = −1
10) 4𝑥𝑦𝑧 𝑤ℎ𝑒𝑛 𝑥 = 2, 𝑦 = 3, 𝑧 = −4
11) −2𝑥2𝑦𝑧 𝑤ℎ𝑒𝑛 𝑥 = 1, 𝑦 = −2, 𝑧 = 4
12) 6𝑥2𝑦𝑧2 𝑤ℎ𝑒𝑛 𝑥 = 2, 𝑦 = 4, 𝑧 = 1
13) 3𝑥 + 4𝑦 𝑤ℎ𝑒𝑛 𝑥 = 5, 𝑦 = −6
14) 5𝑥 − 2𝑦 𝑤ℎ𝑒𝑛 𝑥 = 4, 𝑦 = −3
15) 2𝑥 − 3𝑦 + 7𝑧 𝑤ℎ𝑒𝑛 𝑥 = 3, 𝑦 = 2, 𝑧 = −4
(𝑥2−2𝑦2) 16) 𝑤ℎ𝑒𝑛 𝑥 = 4, 𝑦 = 3
2𝑥
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(𝑥3−3𝑦2+2) 17) 𝑤ℎ𝑒𝑛 𝑥 = 4, 𝑦 = 2
3𝑦
(4−𝑥2) 18) 𝑤ℎ𝑒𝑛 𝑥 = 8, 𝑦 = 5
4𝑦
19) 2𝑥2 − 3𝑥𝑦 + 𝑦2 𝑤ℎ𝑒𝑛 𝑥 = 4, 𝑦 = 2
20) 4𝑥2 − 2𝑥𝑦 − 𝑦2 𝑤ℎ𝑒𝑛 𝑥 = −2, 𝑦 = 3
21) 2𝑎5𝑏2 𝑤ℎ𝑒𝑛 𝑎 = −1, 𝑏 = −2
(4−𝑥2) 22) 𝑤ℎ𝑒𝑛 𝑎 = 5, 𝑏 = 2
4𝑦
(2𝑎𝑏−𝑏2+1) 23) 𝑤ℎ𝑒𝑛 𝑎 = 4, 𝑏 = 1
2𝑏
(4𝑎𝑏+𝑎+10) 24) 𝑤ℎ𝑒𝑛 𝑎 = 5, 𝑏 = 2
−𝑎
25) 2𝑎3𝑏2 − 4𝑎3 𝑤ℎ𝑒𝑛 𝑎 = −1, 𝑏 = −2
26) 3𝑥2 + 4𝑦3 + 5𝑧4 𝑤ℎ𝑒𝑛 𝑥 = 3, 𝑦 = 2, 𝑧 = 1
27) 2𝑥 − 3𝑦 + 4𝑧3 𝑤ℎ𝑒𝑛 𝑥 = 5, 𝑦 = −2, 𝑧 = −1
28)5𝑎 − 4𝑏 + 3𝑐 − 2𝑑 𝑤ℎ𝑒𝑛 𝑎 = −8, 𝑏 = −6, 𝑐 = −4, 𝑑 = −2
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Section 3.2: Functional Notation
“There’s no such thing as talent; you just have to work hard enough.”
Evaluate:
1) 𝑓(5); 𝑓(𝑥) = 4𝑥 − 3
2) 𝑓(4); 𝑓(𝑥) = 6 − 8𝑥
3) 𝑓(−2); 𝑓(𝑥) = 5 − 𝑥2
4) 𝑔(3); 𝑔(𝑥) = 𝑥2 − 2𝑥 + 6
5) 𝑔(−1); 𝑔(𝑥) = 𝑥2 + 4𝑥 − 9
6) ℎ(−2); ℎ(𝑥) = 3𝑥2 − 4𝑥 − 5
7) 𝐹(−3); 𝐹(𝑥) = −2𝑥2 + 5𝑥 − 6
8) 𝐺(4); 𝐺(𝑥) = −𝑥2 − 7𝑥 + 2
9) 𝐻(3); 𝐻(𝑥) = −4𝑥2 + 3𝑥 − 7
10) ℎ(2); ℎ(𝑥) = 𝑥3 + 𝑥2 − 2𝑥 + 6
11) 𝑓(1); 𝑓(𝑥) = −5𝑥3 + 8𝑥2 − 3𝑥 + 4
12) 𝑓(−1); 𝑓(𝑥) = 2𝑥4 + 4𝑥3 − 6𝑥2 + 9𝑥 − 10
13) 𝐹 (1) ; 𝐹(𝑥) = 10𝑥 + 4
2
14) 𝐹 (1) ; 𝐹(𝑥) = 6𝑥 − 1
3
(𝑥+3) 15) ℎ(3); ℎ(𝑥) =
(𝑥−2)
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(𝑥2+1) 16) 𝑓(3); 𝑓(𝑥) =
𝑥
(𝑥2+2𝑥+3) 17) 𝑔(1); 𝑔(𝑥) =
2𝑥
18) 𝐹(−2); 𝐹(𝑥) = (𝑥 + 4)(𝑥 + 7)
19) 𝐹(1); 𝐹(𝑥) = (𝑥 − 1)(𝑥 + 2)(𝑥 − 3)(𝑥 + 4)
20) 𝑓(2); 𝑓(𝑥) = 𝑥4 − 1
21) 𝐺(1); 𝐺(𝑥) = 𝑥3 + 𝑥2 + 4𝑥 + 8
22) 𝐺(−1); 𝐺(𝑥) = 𝑥3 − 2𝑥2 + 3𝑥 − 7
23) 𝑓(0); 𝑓(𝑥) = 𝑥9 + 3𝑥7 − 5𝑥2 + 2
24) 𝑓(𝑎); 𝑓(𝑥) = 6𝑥 + 2
25) 𝑓(𝑎 + ℎ); 𝑓(𝑥) = 4𝑥 + 3
26) 𝐹(𝑎); 𝐹(𝑥) = 𝑥2 + 8𝑥
27) 𝐹(𝑎 + ℎ); 𝐹(𝑥) = 𝑥2 + 4𝑥 + 6
28) 𝐺(𝑎); 𝐺(𝑥) = 2𝑥2 − 3𝑥 − 6
29) 𝑓(𝑎); 𝑓(𝑥) = −4𝑥2 + 2𝑥 − 6
30) 𝑓(𝑎 + ℎ); 𝑓(𝑥) = −3𝑥2 − 6𝑥 + 8
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Section 4.1: Multiply Monomial∙Monomial
“I will guard everything within the limits of my post and quit my post only when properly relieved.”
Multiply: 1) 𝑥2 ∙ 𝑥3
2) 𝑦3 ∙ 𝑦4
3) 4𝑎2 ∙ 5𝑎6
4) (−4𝑝4)(8𝑝3) 5) (3𝑠5)(−6𝑠7) 6) (−2𝑡2)(−5𝑡3) 7) (3𝑎𝑏2)(8𝑎2𝑏) 8) (4𝑐2𝑑3)(−6𝑐𝑑2) 9) (−2𝑟4𝑠)(5𝑟2𝑠3) 10) (−6𝑥5𝑦3)(−8𝑥2𝑦4) 11) (3𝑥4𝑦5)2
12) (−4𝑎4𝑏6)2
13) 4𝑥2(4𝑥)2
14) (𝑥𝑦2)(𝑥𝑦)2
15) (9𝑥𝑦2)(2𝑥2𝑦)(3𝑥𝑦) 16) (4𝑎𝑏3)(2𝑎2𝑏4)(−3𝑎5𝑏2) 17) (6𝑟𝑠5)(−3𝑟4𝑠3)(−5𝑟2𝑠4) 18) (−5𝑥𝑦2)(−2𝑥3𝑦4)(−8𝑥6𝑦5) 19) (15𝑥2𝑦3𝑧4)(4𝑥𝑦2𝑧5) 20) (−7𝑥2𝑦3𝑧)(9𝑥3𝑦8𝑧10) 21) (−8𝑎2𝑏6)(−3𝑎3𝑏8) 22) (4𝑟2𝑠3𝑡5)(4𝑟𝑠2𝑡4)(3𝑟3𝑠𝑡6) 23) (−2𝑥2𝑦3𝑧)(3𝑥𝑦4𝑧2)(4𝑥2𝑦2𝑧5) 24) (8𝑎2𝑏2𝑐5)(−4𝑎3𝑏4𝑐7)(−3𝑎𝑏5𝑐6) 25) (−2𝑝5𝑞4𝑟3)(−3𝑝4𝑞3𝑟6)(−5𝑝2𝑞5𝑟6) 26) 3𝑥𝑦(4𝑥𝑦)2
27) −2𝑎𝑏2(3𝑎2𝑏3)2
28) −4𝑥(2𝑥2𝑦3)2(5𝑥2𝑦) 29) (2𝑥𝑦)(3𝑥2𝑦)2(10𝑥2𝑦) 30) (3𝑥2𝑦)2(2𝑥3𝑦4)2
31) (2𝑥3)(4𝑥2)(−5𝑥)
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Section 4.2: Multiply Monomial∙Binomial-The Distributive Law
“In theory, there’s no difference between theory and practice, but in practice there is.”
Multiply:
1) 8(2𝑥 + 6) 2) −5(4𝑥 + 10) 3) 6𝑥(3𝑥2 + 7) 4) 𝑥2(𝑥 + 4) 5) −2𝑦2(𝑦2 + 3) 6) 4𝑦3(2𝑦4 − 6𝑦) 7) −9𝑧2(3𝑧2 − 5𝑧) 8) −4𝑧3(8𝑧2 + 2𝑧) 9) (2𝑎 + 5)𝑎3
10) −5𝑎2(3𝑎 − 1) 11) 4𝑏(11𝑏 − 2𝑏2) 12) 6𝑐(𝑐2 + 4𝑐 − 8) 13) −2𝑟(𝑟2 − 10𝑟 + 9) 14) −6𝑠2(4𝑠2 + 2𝑠 − 3) 15) 𝑥2𝑦(3𝑥2𝑦-4x𝑦2 + 9) 16) 2𝑎2𝑏3(4𝑎2𝑏 − 5𝑎𝑏 − 𝑎) 17) (𝑥2 + 5𝑥 − 6)(−2𝑥2) 18) 2𝑥(𝑥3 + 4𝑥2 − 10𝑥 + 7) 19) −4𝑟4𝑠3(5𝑟3𝑠2 + 6𝑟2𝑠 + 4𝑟2 − 1) 20) 3𝑥3(5𝑥4 + 3𝑥3 − 2𝑥2 + 9𝑥 − 10)
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Section 4.3: Binomial∙Binomial
“Be kind, for everyone you know is fighting a hard battle.”
Multiply: 1) (𝑥 + 1)(𝑥 + 3) 2) (𝑥 − 2)(𝑥 + 4) 3) (𝑥 + 7)(𝑥 − 5) 4) (𝑥 − 6)(𝑥 − 8) 5) (8 + 𝑦)(2 + 𝑦) 6) (4 + 𝑦)(9 − 𝑦) 7) (3 − 𝑦)(2 + 𝑦) 8) (10 − 𝑦)(4 − 𝑦) 9) (2𝑎 + 3)(𝑎 + 9) 10) (5𝑎 + 6)(2𝑎 − 3) 11) (4𝑏 − 3)(3𝑏 + 5) 12) (8 + 5𝑏)(3 − 2𝑏) 13) (6𝑐 − 5)(2𝑐 − 3) 14) 2(𝑥 + 3)(𝑥 − 9) 15) 5(3𝑥 − 2)(𝑥 − 1) 16) 2𝑥(7𝑥 + 1)(2𝑥 + 3) 17) 3𝑥2(6𝑥 + 1)(𝑥 − 4) 18) (𝑥 + 1)2
19) (𝑥 − 3)2
20) (2𝑥 + 5)2
21) (3𝑥 − 7)2
22) (5 − 3𝑥)2
23) (𝑥 + 4)(𝑥 − 4) 24) (𝑥 + 9)(𝑥 − 9) 25) (6 + 𝑦)(6 − 𝑦) 26) (5 − 𝑧)(5 + 𝑧) 27) (4𝑥 + 5)(4𝑥 − 5) 28) 3𝑥(𝑥 + 2)2
29) 5𝑥(2𝑥 + 1)2
30) 6𝑥2(𝑥 + 4)2
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Section4.4: Multiply Binomial∙Trinomial or Trinomial∙Trinomial
“So much that happens, happens in small ways.”
Multiply:
1) (𝑥 + 1)(𝑥2 + 2𝑥 + 3) 17) (𝑥2 + 1)(𝑥2 + 5𝑥 + 7)
2) (𝑥 + 2)(𝑥2 − 5𝑥 + 6) 18) (𝑥2 + 2𝑥 + 1)(𝑥2 − 4𝑥 + 4)
3) (𝑥 − 3)(𝑥2 − 2𝑥 + 8) 19) 2𝑥(𝑥 + 5)(𝑥2 + 4𝑥 − 12)
4) (𝑦 + 8)(𝑦2 + 4𝑦 + 4) 20) (𝑥 + 1)2(𝑥 − 2)2
5) (𝑦 − 4)(2𝑦2 + 5𝑦 − 12)
6) (3𝑦 − 5)(𝑦2 + 7𝑦 + 10)
7) (5𝑧 − 6)(𝑧2 + 𝑧 + 8)
8) (7𝑧 + 3)(3𝑧2 − 13𝑧 − 5)
9) (𝑧 + 9)(4𝑧3 + 6𝑧 − 3)
10) (𝑎 + 10)(5𝑎3 − 4𝑎2 + 2𝑎)
11) (𝑎 + 7)(𝑎 + 1)2
12) (𝑎 − 3)(𝑎 + 2)2
13) (𝑏 + 1)3
14) (𝑐 + 2)3
15) (2𝑟 + 1)3
16) (3𝑠 − 2)3
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Section 5.1: Combining Like Terms
Combine Like Terms: 1) 4𝑥 − 6𝑥 − 7𝑥 2) 5𝑥𝑦 + 8𝑥𝑦 − 4𝑥𝑦 3) 7𝑥2𝑦 − 2𝑥2𝑦 + 3𝑥2𝑦 4) 6(𝑥 + 4) − 5𝑥 5) 9(𝑥 + 𝑦) − 7(𝑥 − 𝑦) 6) 8(2𝑎 + 3𝑏) − 4(5𝑎 − 2𝑏) 7) 3(8𝑝 − 2) − 2(4𝑝 − 1) + 7(𝑝 − 4) 8) 10𝑝𝑞 + 5𝑝2𝑞 − 4𝑝𝑞 9) 11𝑟𝑠2 + 4 − 2𝑟𝑠2 + 6 10) 4𝑟𝑠2 + 6𝑟2𝑠 − 2𝑟𝑠 11) 2𝑥𝑦(3𝑥 + 5𝑦) − 4𝑥(𝑥𝑦 + 2𝑦) 12) 6𝑟2𝑠 − 2𝑟(𝑟𝑠2 + 5𝑟𝑠) − 6𝑟𝑠2
13) 4𝑎3 − 5𝑎(3𝑎2 + 2𝑏) − 7𝑎(2𝑏 + 1) 14) 2(𝑥 + 4) − 𝑥(𝑥 − 3) + 10𝑥2
15) (2𝑥2 + 3𝑥 − 4) + (5𝑥2 − 8𝑥 − 2) − (3𝑥2 − 6𝑥 − 1) 16) (8𝑦2 − 5𝑦 + 3) + (2𝑦2 + 6𝑦 − 9) − (𝑦2 − 7𝑦 − 11) 17) 3𝑥2(𝑥 − 4) − 𝑥(3𝑥 − 5) − 16 18) −5𝑠𝑡(2𝑠2𝑡2 − 3𝑠𝑡) + 10𝑠3𝑡3
19) 5𝑥(𝑥 + 4) − 2(𝑥2 − 𝑥) − 3(𝑥 + 9) 20) 3𝑦(𝑦 + 1) − 6(𝑦2 + 𝑦) − (𝑦 − 7) − 4 21) −3𝑐𝑑(2𝑐2 − 3𝑐𝑑 + 4𝑐𝑑2) + 5𝑐2𝑑 22) 5𝑎 − (3𝑎 − 𝑏) − 𝑏 23) 9 − 7(2𝑥 − 𝑦) + 7𝑦 24) (2𝑥2𝑦 + 5𝑦) − (3𝑥2𝑦 − 2𝑦) 25) (8𝑥2 − 3𝑥 − 9) − (4𝑥2 + 2𝑥 − 7) 26) 4𝑥2(2𝑥 − 3) − 5𝑥(𝑥2 + 6𝑥) 27) −4𝑟𝑠(6𝑟2𝑠2 − 5𝑟𝑠) − 10𝑟2𝑠2(𝑟𝑠 + 2) + 10𝑟𝑠 28) 2𝑎𝑏2 − 3𝑎(𝑎𝑏2 + 5𝑎2𝑏) + 9𝑎2𝑏2
29) 6𝑝2𝑞 − 2𝑝(𝑝𝑞2 − 5𝑝𝑞) − 7𝑝2𝑞2
30) 8 − 2(5𝑥2 − 4) − 3(2𝑥2 − 1) − (𝑥2 − 5) 31) 5𝑥(2𝑥2𝑦2 − 4) − 2(𝑥3𝑦 + 3𝑥) + 2𝑥2
32) 2𝑥6(4𝑥3 − 3𝑦) − 4𝑥(𝑥8 − 2𝑥5𝑦 + 5(𝑥9 + 𝑦) 33) (8𝑦2 − 4𝑦 − 3) − (2𝑦2 − 3𝑦 − 7) − (6𝑦2 + 𝑦 − 4) 34) 3(4𝑧2 − 2𝑧 − 1) − 5(3𝑧2 − 4𝑧 − 2) + 2(𝑧2 − 2𝑧 − 9) 35) 6 − 5(𝑥2 + 4𝑥 − 3) − 3(2𝑥2 − 5𝑥 − 2) − 9(3𝑥2 − 6𝑥 + 8)
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Section 6.1: Unique Prime Factorization of Integers
“So we beat on, boats against the current, borne back ceaselessly into the past.”
Factor into prime factors:
1) 12 29) 360
2) 24 30) 1440
3) 96
4) 144
5) 18
6) 98
7) 20
8) 200
9) 72
10) 36 11) 99
12) 120
13) 48
14) 54
15) 80
16) 500
17) 108
18) 40
19) 42
20) 75
21) 27
22) 88
23) 480
24) 105
25) 250
26) 245
27) 720
28) 880
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Section 6.2: Greatest Common Monomial Factor (GCF)-Numerical or Algebraic
“As my pappy always used to say: there ain’t no horse that can’t be broken and there ain’t no rider that can’t be throwed.”
Factor Completely:
1) 2x – 4 2) 8y + 16 3) 15𝑧2 + 8𝑧 4) 3𝑥2 + 6𝑥 + 12 5) 4𝑦3 + 20𝑦2 − 8𝑦 6) 𝑎4 + 𝑎3 − 𝑎2 + 5𝑎 7) 7𝑏3 − 14𝑏2 + 21𝑏 8) 4𝑎2𝑏2 − 2𝑎𝑏 + 6𝑎2𝑏 − 8𝑎𝑏2
9) 36x2y3 + 12x𝑦3
10) 36𝑎3𝑏4 + 18𝑎2𝑏3 + 12𝑎𝑏2 + 6𝑎𝑏 11) 14r3 − 28𝑟2 + 35𝑟 12) 𝑠6 + 2𝑠4 − 𝑠2
13) 20𝑎10 + 10𝑎5 − 5𝑎3
14) 2𝑡3 − 4𝑡2 + 8𝑡 15) 18𝑥3𝑦4𝑧5 − 27𝑥2𝑦3𝑧3 + 9𝑥2𝑦2𝑧2
16) 12𝑦7 − 9𝑦5 + 3𝑦2
17) 11𝑎2𝑏𝑐 − 22𝑎2𝑏2𝑐 + 33𝑎2𝑏2𝑐2
18) 2𝑥2 − 3𝑦2 − 4𝑧2
19) 2𝑥𝑦𝑧2 − 3𝑥𝑦𝑧 + 4𝑦𝑧 20) 4𝑎𝑏𝑐 + 5𝑎𝑏 − 6𝑏𝑐 21) 4𝑥3 + 8 22) 12𝑐4 + 48𝑐3 − 96𝑐2
23) 15𝑥𝑦2 + 30𝑥𝑦3
24) 10𝑎2𝑏 + 20𝑎𝑏2 − 30𝑎𝑏 25) 6𝑥2 + 16𝑦2 + 26𝑧2
26) 36𝑏4 + 72𝑐4
27) 8𝑥3𝑦4 + 16𝑥2𝑦3 + 4𝑥𝑦 28) 24𝑏4 + 12𝑏3 + 6𝑏2
29) 40α +20αβ +10αβµ 30) 9𝜃2 + 18𝜃
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Section 6.3: The Difference of Two Squares (DOTS)
“I know it happens to everyone, but it never happened to me.”
Factor Completely:
1) 𝑥2 – 9 30) 36 𝑔4ℎ10 − 49
2) 𝑥2 – 36 31) 𝑎16𝑏36 −36
3) 𝑦2 – 64 32) 4𝑥2 − 9
4) 𝑦2 – 81 33) 𝑚4 − 𝑛4
5) 100 – 𝑧2 34) (𝑟𝑠)2 − 25
6) 25 – 𝑎2 35) 𝑔6ℎ4 − 25
7) 𝑏2 + 16
8) 𝑥2 + 49
9) 𝑟2 – 10
10) 𝑥2 – 19
11) 𝑥4 – 1
12) 𝑥4 – 16
13) 𝑦8 – 144
14) 16𝑥16– 49
15) 36𝑦16– 121
16) 𝑏10 – 𝑎2
17) 16𝑥16– 49
18) 𝑐2– 𝑑2
19) 𝑔6ℎ4 −𝑚2
20) 1 − 4𝑥2
21) 9 − 25𝑦2
22) 16𝑥2 − 49𝑦2
23) 81𝑥2 − 4𝑧2
24) 64𝑦2 − 121𝑤2
25) 𝑧4 − 𝑦2
26) 36𝑟2𝑠4 − 𝑡2
27) 25𝑎6𝑏8 − 4
28) 𝑥36 − 4
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Section 6.4: Combination of Greatest Common Monomial Factor and Difference of Two Squares
“My grandfather said that this reminded him of his grandfather, now this
Reminds me of my grandfather”
Factor Completely:
1) 2𝑥2 − 2 2) 3𝑦2 − 27 3) 6𝑧 − 24𝑧3
4) 4𝑎2 − 36 5) 2 − 8𝑏2
6) 27− 75𝑐2
7) 4𝑑2 − 100 8) 10𝑥4 − 10 9) 2𝑦4 − 32 10) 8𝑧2 − 18 11) 72𝑎2 − 8 12) 36𝑏36 − 36 13) 16− 16𝑐16
14) 𝑟4𝑠2 − 𝑠2
15) 𝑥4 − 121𝑥2
16) 3𝑦3 − 75𝑦 17) 4𝑎2𝑏2 − 𝑎4
18) 6𝑎𝑏𝑐2 − 24𝑎𝑏 19) 18𝑥2𝑦3𝑧4 − 2𝑥2𝑦3𝑧2
20) 4𝑥4𝑦2 − 36𝑦2
21) 𝑎5 − 𝑎3
22) 4𝑏4 − 𝑏2
23) 25𝑥4 − 4𝑥2
24) 5𝑥𝑦2 − 125𝑥 25) 9𝑥2 − 81 26) 4𝑥3 − 16𝑥 27) 𝑎3 − 144𝑎 28) 7𝑏3 − 63𝑏 29) 9𝑐3𝑑3 − 36𝑐𝑑 30) 5 − 5𝑟2
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29
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32
33
34
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36
37
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39
40
Section 6.5: Trinomial With Leading Coefficient One ( x2 + 𝑏𝑥 + 𝑐) Factor Completely:
1) 𝑥2 + 3𝑥 + 2
2) 𝑥2 − 3𝑥 + 2
3) 𝑥2 + 𝑥 − 2
4) 𝑥2 − 𝑥 − 2
5) 𝑦2 + 5𝑦 + 6
6) 𝑦2 + 𝑦 − 6
7) 𝑦2 − 𝑦 − 6
8) 𝑦2 − 5𝑦 − 6
9) 𝑧2 + 4𝑧 + 3
10) 𝑧2 − 4𝑧 + 3
11) 𝑧2 − 2𝑧 − 3
12) 𝑧2 + 2𝑧 − 3
13) 𝑎2 +10a +24
14) 𝑏2 + 7𝑏 − 30
15) 𝑐2 − 6𝑐 + 8
16) 𝑟2 − 13𝑟 + 30
17) 𝑠2 − 9𝑠 + 18
18) 𝑡2 − 4𝑡 − 12
19) 𝑤2 + 2𝑤 − 15
20) 𝑑2 − 11𝑑 + 28
) 𝑥2 − 2𝑥 − 48
) 𝑦2 − 3𝑦 − 54
) 𝑧2 − 17𝑧 + 30
) 𝑎2 + 4𝑎 − 60
) 𝑏2 + 10𝑏 + 25
) 𝑐2 + 5𝑐 − 14
) 𝑑2 − 16𝑑 + 28
) 𝑥2 − 8𝑥 − 20
) 𝑦2 + 5𝑦 − 66
) 𝑣2 + 6𝑣 − 55
) 𝑢2 + 6𝑢 + 5
) 𝑎2 − 3𝑎 − 10
) 𝑏2 − 6𝑏 − 16
) 𝑐2 + 7𝑐 − 18
) 𝑑2 + 9𝑑 + 20
) 𝑥2 − 5𝑥 − 36
) 𝑦2 + 14𝑦 + 40
) 𝑧2 + 6𝑧 + 5
) 𝑎2 + 14𝑎 + 48
) 𝑏2 + 𝑏 − 72
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Section 6.6: Trinomials with Leading Coefficients Other Than One
(𝑎𝑥2 + 𝑏𝑥 + 𝑐) Factor Completely:
1) 2𝑥2 + 5𝑥 + 2 2) 2𝑥2 − 5𝑥 + 2 3) 2𝑥2 − 3𝑥 − 2 4) 2𝑥2 + 3𝑥 − 2 5) 3𝑦2 − 7𝑦 + 2 6) 3𝑦2 + 7𝑦 + 2 7) 3𝑦2 − 5𝑦 + 2 8) 3𝑦2 + 5𝑦 + 2 9) 5𝑧2 − 14𝑧 − 3
10) 5𝑧2 + 14𝑧 − 3
11) 5𝑧2 − 16𝑧 + 3
12) 5𝑧2 + 16𝑧 + 3
13) 3𝑎2 − 𝑎 − 2
14) 5𝑏2 + 2𝑏 − 3
15) 3𝑐2 − 7𝑐 + 4
16) 4𝑑2 + 5𝑑 − 6
17) 4𝑔2 + 10𝑔 − 24
18) 6𝑟2 + 11𝑟 + 5
19) 6𝑠2 − 7𝑠 − 5
20) 8𝑡2 − 6𝑡 − 9
21) 8𝑢2 + 18𝑢 + 9
22) 9𝑣2 + 12𝑣 + 4
23) 9𝑤2 + 6𝑤 − 8
24) 10𝑥2 − 19𝑥 + 6
25) 12𝑦2 + 8𝑦 − 15
26) 14𝑧2 − 19𝑧 − 3
27) 15𝑎2 + 𝑎 − 6
28) 16𝑏2 − 4𝑏 + 25
29) 16𝑐2 + 8𝑐 + 1
30) 20𝑑2 + 3𝑑 − 2
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Section 6.7: Combination of Greatest Common Monomial. Factor and Trinomial
Factor Completely:
1) 4𝑥2 − 4𝑥 − 8
2) 2𝑥3 + 4𝑥2 + 2𝑥
3) 𝑧5 − 2𝑧4 − 3𝑧3
4) 5𝑎2 − 40𝑎 − 45
5) 6𝑏2 − 6𝑏 − 120
6) 2𝑐2 − 6𝑐 − 20
7) 4𝑏2 + 20𝑏 + 24
8) 3𝑟3 − 3𝑟2 − 18𝑟
9) 5𝑠4 − 30𝑠3 − 35𝑠2
10) 2𝑡3 + 26𝑡2 − 60𝑡
11) 4𝑢4 + 12𝑢3 + 8𝑢2
12) 4𝑣2 + 16𝑣 + 12
13) 6𝑥2 + 30𝑥 + 36
14) 8𝑤3 + 24𝑤2 + 16𝑤
15) 8𝑦2 + 32𝑦 + 24
16) 10𝑧2 + 70𝑧 − 300
17) 11𝑎2 − 66𝑎+88
18) 12𝑏4 + 24𝑏3 − 180𝑏2
19) 15𝑐2 − 45𝑐 − 150
20) 20𝑑2 − 160𝑑 − 400
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Section 6.8: Factoring by Grouping
“The person who does the work is the only one who learns.”
Factor by Grouping
1) 𝑥(𝑥 + 4) + 2(𝑥 + 4)
3) 𝑧(𝑧 + 9) − 7(𝑧 + 9)
5) 4𝑏(𝑏 + 7) + 3(𝑏 + 7)
7) 7𝑑(𝑑 − 8) − 5(𝑑 − 8)
9) 𝑥3 − 6𝑥2 + 4𝑥 − 24
11) 𝑧3 − 4𝑧2 + 3𝑧 − 12
13) 2𝑥2 + 4𝑥 + 3𝑥 + 6
15) 4𝑧2 − 𝑧 − 8𝑧 + 2
17) 3𝑥3 + 9𝑥2𝑦 − 4𝑥 − 12𝑦
19) 5𝑎2𝑏+15𝑎2𝑐 − 6𝑏 − 18𝑐
21) 63𝑢2𝑣 + 18𝑢2𝑤 + 35𝑣𝑥 + 10𝑤𝑥
2) 𝑦(𝑦 − 6) + 5(𝑦 − 6)
4) 𝑎(𝑎 − 3) − 8(𝑎 − 3)
6) 5𝑐(𝑐 + 11) − 8(𝑐 + 11)
8) 12𝑟(𝑟 − 1) − 17(𝑟 − 1)
10) 𝑦3 + 5𝑦2 − 6𝑦 − 30
12) 𝑎4 + 3𝑎3 − 6𝑎 − 18
14) 2𝑦2 − 3𝑦 − 14𝑦 + 21
16) 𝑎2 − 6𝑎 − 6𝑎 + 36
18) 4𝑥𝑦2 + 20𝑦2𝑧 + 3𝑥 + 15𝑧
20) 21𝑟𝑠 − 35𝑟𝑡 + 12𝑠𝑣 − 20𝑡𝑣
22) 12𝑥𝑦 + 15𝑥𝑧 + 20𝑥𝑟 + 25𝑧𝑟
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Section 6.9: Review of All Factoring
Factor Completely: 1) 36−𝑥2
2) 𝑥2 − 7𝑥 − 60 3) 28𝑎2𝑏3 − 35𝑎2𝑏2 + 7𝑎𝑏 4) 3𝑦2 − 12𝑦 − 15 5) 𝑥4 − 1 6) 8𝑦3𝑧3 − 32𝑦𝑧 7) 16𝑎16 − 25 8) 3𝑏2 − 5𝑏 − 2 9) 4𝑥𝑦𝑧 + 5𝑥𝑦 + 6𝑦𝑧 10) 9 − 25𝑎2
11) 𝑏4𝑐2 − 𝑐2
12) 𝑑2 − 4𝑑 + 3 13) 𝑟2 + 7𝑟 − 30 14) 9𝑠2 + 12𝑠 + 4 15) 𝑡2 + 12𝑡 + 10 16) 9𝑢3 − 27𝑢2 − 90𝑢 17) 15𝑣3 + 12𝑣2
18) 12𝑥4𝑦 − 8𝑥3𝑦 − 4𝑥2𝑦 + 4𝑥𝑦 19) 4𝑥2 − 4𝑥 20) 42 − 13𝑥 + 𝑥2
21) 20𝑥2𝑦 − 25𝑥𝑦2 + 30𝑥𝑦 22) 6𝑧2 − 7𝑧 − 5 23) 𝑥4 − 16 24) 8𝑦2 − 18 25) 2𝑧2 − 7𝑧 − 15 26) 2𝑎3 − 20𝑎2 + 48𝑎 27) 21 + 10𝑏 + 𝑏2
28) 40𝑎3𝑏2 − 20𝑎2𝑏 + 10𝑎𝑏2 + 5𝑎𝑏 29) 𝑥2 − 25 30) 6𝑥2 − 5𝑥 − 4 31) 6𝑥2 + 6𝑥 − 5𝑥 − 5 32) 4𝑦2 + 16𝑦 + 3𝑦 + 12 33) 6𝑥𝑧2 + 9𝑦𝑧2 + 10𝑥 + 15𝑦 34) 15𝑎2𝑏 − 20𝑎2𝑐 − 21𝑏 + 28𝑐
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Section 7.1: Is a Given Number a Solution to an Equation
“In mathematics, we never understand things, but just get used to them.”
Determine whether or not the given number is a solution to the equation 1) 𝑥 = −1, and 15𝑥 + 20 = 5
2) 𝑦 = −2, and 12 − 5𝑦 = 2
3) 𝑧 = 3, and 10 − 4𝑧 = 1 − 𝑧
4) 𝑎 = 4, and 8 − 4𝑎 + 2 = 3(2 − 𝑎)
5) 𝑏 = 2, and 3(𝑏 − 5) + 6 = 2(𝑏 + 1)
6) 𝑐 = −3, and 𝑐 − 4(𝑐 + 5) = 4 − 2(1 − 𝑐)
7) 𝑑 = −1, and 𝑑2 − 4𝑑 + 3 = 0
8) 𝑟 = 6, and 𝑟2 + 10𝑟 + 24 = 0
9) 𝑠 = −2, and 𝑠2 = 𝑠 + 2
10) 𝑡 = 2, and (𝑡 − 2)(𝑡 + 5) = 0
11) 𝑤 = −3, and (𝑤 − 3)(𝑤 + 1) = 0
12) 𝑥 = 8, and (𝑥 − 8)(𝑥 + 4)(𝑥 + 3)(𝑥 + 2) = 0
13) 𝑦 = −1, and 2𝑦3 − 4𝑦2 + 𝑦 − 9 = 0
14) 𝑧 = 1, and 2𝑧4 − 3𝑧3 + 4𝑧2 + 5𝑧 − 6 = 0
15) 𝑎 = −7, and 9 − (𝑎 − 3) + 4𝑎 = 5𝑎 − 5(𝑎 + 2) + 6
16) 𝑏 = 4, and 3(2𝑏 − 3) − 4(3𝑏 − 1) + 5 = 6(4 − 2𝑏)
17) 𝑐 = 1, and 9(2𝑐 + 3) − 8𝑐 = 5(3𝑐 + 1) + 16
18) 𝑥 = −3, and 2(1 − 2𝑥) + 3(4𝑥 + 5) = 4 − (1 − 4𝑥)
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Section 7.2: Linear (First Degree) Equations
“Touching the dull formulas with his wand, he turned them into poetry.”
Solve each equation: 1) 4𝑥 = 36 2) 6𝑥 = 72 3) −5𝑥 = 20 4) −8𝑥 = 48 5) −2𝑦 + 1 = 15 6) 2𝑦 + 4𝑦 = 54 7) 𝑦 + 3𝑦 + 5𝑦 = 81 8) 4𝑦 + 6𝑦 + 2𝑦 = 60 9) 3𝑧 + 4 = 31 10) 5𝑧 − 9 = 41 11) 2𝑎 + 7 = 3𝑎 − 9 12) 3𝑎 + 7 = 8𝑎 − 18 13) 5𝑏 + 11 = 7𝑏 − 3 14) 2𝑥 + 3𝑥 − 9 = 4𝑥 + 5𝑥 + 3 15) 5 + 3(𝑥 + 1) = 8 16) 2(𝑐 + 4) = 3(𝑐 − 5) 17) −4(𝑐 + 3) = 5(𝑐 + 3) 18) 8(𝑑 + 4) = 5𝑑 − 28 19) 20 − 5𝑑 = −2(𝑑 − 1) 20) 10 − 9𝑟 = 8 − 3(𝑟 + 2) 21) 𝑟 − 6(𝑟 − 4) = 3 − (3 − 7𝑟) 22) 5 − 2(𝑠 + 3) + 20 = 5(2𝑠 − 1) 23) 3𝑠 − 5(1 − 𝑠) = 3(𝑠 + 1) + 2 24) 6 − 2(𝑡 − 5) = 12 + 3(2𝑡 − 4) 25) 3(𝑡 − 2) − 2 = 5(𝑡 + 3) − 7(𝑡 − 1) 26) (𝑢 − 3) − 2(𝑢 − 4) = 5(𝑢 − 6) − 𝑢 27) 2(𝑣 − 3) − (𝑣 − 4) = 4(3 − 𝑣) 28) 2(𝑥 + 1) + 4(2𝑥 − 3) = 3 − (4𝑥 + 1) 29) 40𝑥 + 60(𝑥 − 1) = 40 + 20(𝑥 − 1) 30) 5 − (𝑦 − 3) + 3𝑦 = 4𝑦 − 5(𝑦 + 3) − 1 31) 5 − 2(𝑧 + 3) + 20 = 5(2𝑧 − 1) 32) 2(3𝑥 + 1) + 4(5𝑥 + 2) + 6(7𝑥 + 9) = 64 33) 20(2𝑧 + 3) + 40(3𝑧 + 4) + 60(8𝑧 + 1) = 920 34) 2(𝑧 + 3) + 5(𝑧 + 4) = 6(2𝑧 + 9) + 8(3𝑧 − 4)
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Section 7.3: Decimal Equations
“Whatever we invent in mathematics seems independent of our inventing.” “Mathematics lies in an enchanted world, somewhere between reality and
imagination.”
Solve for each variable: 1) . 4𝑥 = .8 2) . 6𝑥 = 1.8 3) 2.4𝑥 + 1.2𝑥 = 7.2 4) . 3𝑦 + 2 = 1.2 5) 2.4𝑦 + 3 = 7.8 6) 𝑦 + .4 = 5 7) . 02𝑧 = 4 8) . 4𝑧 = .16 9) . 02𝑧 = .7𝑧 + .68 10) 7.4 = 1.8𝑎 − 16 11) . 04𝑎 − 3.2 = .2𝑎 12) . 3𝑎 − 1.2 = .1𝑎 + .4 13) . 48 − .09𝑏 = 1.88 14) 1.5𝑏 + 2 = 1.7𝑏 − 4.6 15) . 5(5𝑐 − 1) − .2(𝑐 + 4) = 1 16) . 25(3𝑐 − 4) − .2 = .25𝑐 + .3 17) 1.6𝑟 + 1.51 = .09𝑟 18) . 48 − .07𝑟 = .89𝑟 19) 1.2𝑠 − 4 = .4𝑠 + 28 20) 3.5𝑠 − 1.5 = 3.14𝑠 + 2.10 21) 3.6𝑡 + 1.73 = 2.86𝑡 − .49 22) . 17𝑤 + 1.56 = .14𝑤 + 1.2 23) 𝑤 + 0.24 = 1.6𝑤 − 0.84 24) 0.02𝑥 + 0.4𝑥 + 2.1 = 0 25) . 08(𝑥 + 1) − .4(𝑥 + 2) = .24 26) . 04𝑥(𝑥 + 2) + .2(𝑥 + 4) = .16 27) . 002𝑦 = 6 28) . 004𝑦 + 3.2 = .02𝑦 29) . 006𝑧 + .04𝑧 + .2𝑧 = .496 30) 𝑥 + .3𝑥 + .03𝑥 + .003𝑥 = 3.9
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Section 7.4: Fractional Equations with Monomials in the Numerator or Binomials in the Numerator
“On earth, the broken arcs; in heaven, a perfect round.”
Solve each equation
𝑥 7𝑥−
5𝑥 𝑥 5 1) = 4 18) = +
5 6 8 3 24
3𝑥 1 3 7 27 2) = 9 19) + + =
7 𝑥 2𝑥 8𝑥 64
−5𝑥 1 5 4 3) = 10 20) + + = 1
8 4𝑥 12𝑥 3𝑥
3 2𝑥 𝑥−1 4) 𝑦 −
14
= 21) = 2 3 4
1 𝑦 2 3 5) + = 2 22) =
9 27 𝑥−4 𝑥−5
𝑦 𝑦 3 3𝑥 𝑥+1 6) + = 23) =
2 4 8 5 2
1 5 𝑥 𝑥−2 −
1 7) = 24) + = 6
𝑧 3𝑧 6 3 5
1 1 2 𝑥+2 𝑥−1 8) + = 25) + = 5
𝑎 15 3 3 6
1 4 3 𝑥−1 2(𝑥+2) −
4 9) + = 26) + = 9
5𝑥 2 5 𝑥 9 3
𝑥 7𝑥 11 𝑥+1 4𝑥−1 2 10) + = 27) + =
3 12 6 5 15 15
2𝑏 7 2(𝑥−5) 3𝑥 𝑥+5 −
1 11) =b + 28) + =
3 2 6 6 5 10
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12) 𝑟 3𝑟 16
+ = 3 5 5
29) 𝑥 2(𝑥−1)
− 𝑥+5
+ = 1 5 9 10
13) 𝑠
− 5𝑠 −1
= 2 9 6
30) 𝑥 𝑥 𝑥 𝑥
+ + + = 15 2 4 8 16
14) 𝑡 2𝑡 23
+ = 9 5 15
15) 𝑥 𝑥 𝑥
+ + =12 12 6 4
16) 12 3 9
+ = 𝑦 2𝑦 4
17) 1 3 2
+ = 2𝑧 8 𝑧
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Section 7.5: Literal Equations
“I’d rather listen to a tree than to a mathematician.”
Solve for the indicated variable:
1) Solve for r: 𝑑 = 𝑟𝑡
2) Solve for l: 𝐴 = 𝑙𝑤
3) Solve for I: 𝑉 = 𝐼𝑅
4) Solve for r: 𝐶 = 2𝜋𝑟
5) Solve for h: 𝐴 = 1
𝑏ℎ 2
6) Solve for h: 𝑉 = 𝑙𝑤ℎ
7) Solve for R: 𝑃𝑉 = 𝑛𝑅𝑇
8) Solve for r: 𝐼 = 𝑝𝑟𝑡
1 9) Solve form: 𝐾 = 𝑚𝑣2
2
10) Solve for r: 𝑎 = 𝑣2⁄𝑟
11) Solve for t: 𝑣2 = 𝑣1 + 𝑎𝑡
12) Solve for a: 𝑃 = 2𝑎 + 2𝑏
13) Solve for b: 𝑃 = 𝑎 + 𝑏 + 𝑐
14) Solve for C: 𝐹 = 9
𝐶 + 32 5
15) Solve for l: 𝑆 = 2𝜋𝑟𝑙 + 2𝜋𝑟2
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2𝜋 5 16) Solve for w: 𝑇 = 28) Solve for F: 𝐶 = (𝐹 − 32)
𝑤 9
𝐺𝑚1𝑚2 17) Solve for 𝑚1: 𝐹 = 29) Solve for z: 2𝑥 + 3𝑦 + 4𝑧 = 10
𝑟2
18) Solve for 𝑏1: 𝐴 = ℎ(𝑏1 + 𝑏2) 30) Solve for y: 𝐴𝑥 + 𝐵𝑦 = 𝐶
19) Solve for r: 𝐴 = 𝑃(1 + 𝑟𝑡)
𝜋𝑟2ℎ 20) Solve for h: 𝑉 =
3
21) Solve for x: 𝑦2 = 4𝑝𝑥
22) Solve for t: 𝑉 = 𝐾 + 𝑔𝑡
𝑚𝑣2
23) Solve for r: 𝐹 = 𝑔𝑟
1 1 1 24) Solve for R: = +
𝑅 𝑅1 𝑅2
𝑃1𝑉1 𝑃2𝑉2 25) Solve for 𝑉2: =
𝑇1 𝑇2
26) Solve for y: 2𝑥 + 3𝑦 = 6
27) Solve for y: 4𝑥 − 5𝑦 = 20
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Section 7.6: Quadratic (Second Degree) Equations
Solve for each variable: 1) (𝑥 − 2)(𝑥 − 5) = 0 36) 50𝑥2 − 25𝑥 = 0 2) (2𝑥 + 1)(𝑥 − 3) = 0 37) 36𝑦2 = 24𝑦 3) 𝑦(𝑦 − 6) = 0 38) 4𝑧2 + 12𝑧 + 8 = 0 4) 𝑦(10 − 𝑦) = 0 39) 6𝑎2 + 11𝑎 + 5 = 0 5) 4𝑧(𝑧 − 9) = 0 40) 𝑏2 + 10𝑏 + 25 = 0 6) 8𝑧(𝑧 + 11) = 0 7) (𝑎 + 8)(𝑎 − 8) = 0 8) (6 − 𝑎)(6 + 𝑎) = 0 9) 𝑥2 − 𝑥 − 12 = 0 10) 𝑥2 + 2𝑥 − 8 = 0 11) 𝑥2 − 8𝑥 − 20 = 0 12) 𝑥2 + 2𝑥 − 15 = 0 13) 2𝑦2 + 11𝑦 + 5 = 0 14) 6𝑦2 + 13𝑦 + 6 = 0 15) 4𝑦2 + 5𝑦 − 6 = 0 16) 10𝑦2 − 19𝑦 + 6 = 0 17) 4𝑧2 − 4𝑧 − 8 = 0 18) 6𝑧2 − 6𝑧 − 120 = 0 19) 𝑧5 − 2𝑧4 − 3𝑧3 = 0 20) 5𝑧4 − 30𝑧3 − 35𝑧2 = 0 21) 𝑎2 − 4𝑎 = 12 22) 𝑏2 + 4𝑏 = 60 23) 𝑐2 = 𝑐 + 2 24) 𝑑2 = 𝑑 + 6 25) 𝑟2 + 8 = 6𝑟 26) 𝑠2 − 48 = 2𝑠 27) 𝑡2 − 4𝑡 = 0 28) 6𝑢2 − 3𝑢 = 0 29) 𝑣2 − 81 = 0 30) 144 − 𝑤2 = 0 31) (𝑥 + 1)(𝑥 + 2) = 0 32) (2𝑥 + 3)((𝑥 + 4) = 0 33) (𝑥 + 5)(𝑥 − 4) + 14 = 0 34) (𝑥 + 6)(𝑥 − 2) − 9 = 0 35) 𝑦2 − 4𝑦 + 4 = 0
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Section 7.7 Linear Inequalities
Solve the following inequalities and sketch the solution on the number line. 1) 2𝑥 > 10 2) 4𝑥 < −12 3) 5𝑥 ≥ 20 4) 8𝑥 ≤ 32 5) −3𝑥 > 15 6) −6𝑥 ≤ 12 7) −10𝑥 ≥ 20 8) −15𝑥 ≤ 30 9) 8𝑥 ≥ −16 10) 9𝑥 ≤ −18 11) −12𝑥 > −24 12) −5𝑥 < −15
𝑥 13) > 3
5
14) 𝑥
≤ 5 9
15) −𝑥
≥ 4 7
16) −𝑥
< −2 3
17) 2𝑥
≥ 8 3
18) 4𝑥
≤ 12 5
−3𝑥 19) > 9
7
20) −4𝑥
< − 8 11
21) 𝑥 + 4 < 6
22) 𝑥 − 5 > −3
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23) 𝑥 + 9 ≤ 2
24) 𝑥 − 7 ≥ −4
25) 4𝑥 − 6 ≥ 10
26) 5𝑥 + 10 < 15
27) 8𝑥 − 7 > 9
28) 6𝑥 + 8 ≤ −4
29) 3𝑥 ≤ 5𝑥 + 8
30) 7𝑥 − 10 < 2𝑥
31) 4𝑥 + 8 > 12𝑥
32) 6𝑥 + 10 ≥ 8𝑥
33) 2𝑥 − 5 > 3𝑥 + 10
34) 4 − 5𝑥 ≤ 𝑥 − 2
35) 3𝑥 + 4 ≥ 5𝑥 − 6
36) 5𝑥 + 4 < 9𝑥 − 8
37) 5𝑥 − (𝑥 + 4) > 12
38) 3(𝑥 + 2) ≤ 3 − 2(𝑥 + 3)
39) 6 − 3(𝑥 + 1) ≥ 6
40) 4 − 5(𝑥 + 2) < 1 − 2(𝑥 − 1)
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Section 8.1: Simplifying Fractions (Rational Expressions)— Monomial/Monomial; Polynomial/Monomial; Polynomial/Polynomial
“There is a thin line that separates a numerator and denominator.”
Simplify:
12 1)
4
−9 2)
36
𝑥3
3) 𝑥
−24𝑦5
4) 3𝑦2
−18𝑥2𝑦 5)
−9𝑥𝑦
48𝑎4𝑏3
6) −16𝑎2𝑏2
−13𝑝4𝑞2
7) 39𝑝6𝑞3
−30𝑟4𝑠 8)
−6𝑟3𝑠
−66𝑎3𝑏7𝑐9
9) 11𝑎2𝑏3𝑐4
−45𝑟6𝑠3𝑡2
10) −9𝑟2𝑠𝑡4
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8𝑥3−4𝑥2+2𝑥 11)
2𝑥
8𝑦𝑧2−4𝑦2𝑧+2𝑦𝑧 12)
−2𝑦𝑧
12𝑎3𝑏2−6𝑎2𝑏+3𝑎𝑏 13)
3𝑎𝑏
24𝑦12+12𝑦6−6𝑦3
14) −6𝑦3
25𝑐25+16𝑐16−9𝑐9
15) −𝑐6
12𝑑7−9𝑑5−3𝑑4
16) −3𝑑4
𝑥−3 17)
3−𝑥
7−𝑦 18)
𝑦−7
4𝑥−4 19)
𝑥2−1
𝑦2−9 20)
3𝑦−9
6𝑧2−12𝑧 21)
𝑧2−4
𝑎2−𝑎−6 22)
𝑎2−9
𝑏2−𝑏−12 23)
𝑏2−16
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6𝑐2−6 24)
𝑐3−𝑐
3𝑟2−3𝑟 25)
𝑟2+5𝑟−6
𝑠2−𝑠−2 26)
𝑠2−2𝑠
𝑡2−𝑡−12 27)
𝑡2+5𝑡+6
𝑥2−9𝑥+20 28)
𝑥2−3𝑥−10
𝑧2−6𝑧+8 29)
2𝑧−8
9𝑥 30)
𝑧2−6𝑧+8
72 𝑥4𝑦6𝑧3
31) 12 𝑥2𝑦3𝑧4
6𝑦2+12𝑦 32)
𝑦2+4𝑦+4
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Section 8.2: Multiplying Fractions (Rational Expressions)
“The winners joke and the losers say deal.”
Multiply:
4∙
12 1)
6 8
−3∙
14 2)
7 6
−25 ∙
−4 3)
2 5
2 ∙
3∙
4∙
5∙
6 4)
3 4 5 6 7
3 5) 14 ∙
7
6) 2𝑥2
∙ 𝑦
𝑦 4𝑥
7) 𝑎3
∙ 𝑏2
𝑏4 𝑎2
8) −4𝑝2
∙ −5𝑞2
15𝑞 12𝑝
9) 30𝑎2
∙ 6𝑏2
18𝑏 5𝑎
10) −6𝑎𝑏𝑐
∙ 10𝑥𝑦2
5𝑥2𝑦 3𝑎𝑏𝑐2
11) −21𝑥2𝑦
∙ −16𝑧2
8𝑧 3𝑥𝑦
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−2𝑥 ∙
6𝑧 ∙
14𝑎 12)
3𝑦 7𝑎 18𝑥
−7𝑥 ∙
−9𝑦 ∙
2𝑦 13)
3𝑦2 14𝑥2 𝑥
14) −44𝑎3
∙ −10𝑏2
∙ −2𝑐2
5𝑏𝑐 4𝑎 11𝑎
15) 3𝑎𝑏𝑐
∙ −8𝑦2𝑧
∙ 6𝑥2
4𝑥𝑦 9𝑎𝑏 15𝑐
(−3𝑞) 16) 25𝑝2𝑞 ∙
10𝑝2
17) −7𝑟2𝑠𝑡3
∙ −12𝑠3
∙ −4𝑟3
6𝑠2 14𝑟𝑡2 𝑡
18) 24𝑎2𝑏3
∙ 10𝑎𝑏
∙ −𝑐3
25𝑎 8𝑎𝑐2 9𝑏2
−6𝑥4𝑦7𝑧9
∙ 30𝑥2𝑦3 39𝑦2
19) ∙ 5𝑥2𝑦 13𝑥𝑦2 36𝑥𝑦𝑧4
20) −2𝑝𝑞𝑟2
∙ −5𝑟3
∙ −6𝑝4
3𝑞3𝑟 4𝑝𝑞2 35𝑟2
𝑥+1 𝑥2−9 21) ∙
2𝑥−6 𝑥2+4𝑥+3
4𝑦3−4𝑦 9 22) ∙
27 𝑦2−1
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Section 8.3: Dividing Fractions (Rational Expressions)
“I was so much older then; I’m younger than that now.”
Divide:
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
2 4 ÷
3 9
−5 10 ÷
6 3
−9 −3 ÷
11 22
𝑥2 𝑥2
÷ 𝑦 𝑦
𝑎4 𝑎2
𝑏3 ÷ 𝑏
5𝑎 𝑎4
÷ 7𝑏 21𝑏2
4𝑥𝑦2 4𝑥𝑦2
3𝑎𝑏2 ÷ 6𝑎𝑏
9𝑝𝑞2 12𝑝4𝑞3
10𝑟𝑠2 ÷ 𝑟𝑠
27𝑐𝑑2 3𝑐𝑑 ÷
10𝑎𝑏 5𝑎𝑏
60𝑠2𝑡 10𝑠𝑡 ÷
7𝑢𝑣 𝑢2
8𝑎2𝑏 −24𝑎𝑏 ÷
27𝑎𝑏 9
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−15𝑥2𝑦2 −5𝑥𝑦2
12) ÷ 36𝑎𝑏 12𝑎𝑏
𝑝5𝑞4𝑟2 𝑝2𝑞3𝑟4
13) ÷ 𝑎𝑏 𝑎4𝑏
−24𝑎6𝑏8𝑐10 −12𝑏4𝑐6
14) ÷ 5𝑎𝑏2 𝑎𝑐
4𝑥2𝑦3𝑧4 −2𝑥𝑦2𝑧4
15) ÷ 5𝑎𝑏𝑐 15𝑎𝑏
25𝑥2𝑦 −5𝑥𝑦 16)
21𝑎4𝑏3 ÷ 𝑎3𝑏2
−72𝑟8𝑠7 9𝑟4𝑠5
17) ÷ 5𝑝𝑞 10𝑝2𝑞2
14𝑥2𝑦𝑧3 −28𝑥𝑦𝑧 18)
11𝑎2𝑏3𝑐4 ÷ 22𝑎𝑏2𝑐3
5𝑥 19) 25𝑥𝑦 ÷
3𝑦
10𝑎2𝑏 20) ÷ 5𝑎𝑏𝑐
7𝑥𝑦
−36𝑥𝑦4 −9𝑥𝑦 21) ÷
11𝑎𝑏 22𝑎
15𝑟𝑠2 5𝑟𝑠 22) ÷
7𝑝𝑞 14
−16𝑝𝑞 −4𝑝𝑞2
23) ÷ 9𝑎2𝑏 3𝑎𝑏
2 4 24) ÷
3𝑥 6𝑥
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𝑥+1 𝑥2−1 25) ÷
𝑥 𝑥2
3𝑥−21 𝑥−7 26) ÷
5 15
𝑥𝑦2 𝑥𝑦 27) ÷
𝑥2−𝑥−12 𝑥2−9
6𝑎2
∙ 3𝑎𝑏 10𝑎3𝑏2
28) ÷ 7𝑏3 5 14𝑎
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Section 8.4: Adding or Subtracting Fractions (Rational Expressions)
“Three out of two people have trouble with fractions.”
Combine:
2 3 4 2 1) + 2) +
5 5 3 3
3 7 −
1 −
2−
1 3) 4)
10 10 9 9 9
1 1 1 1 5) + 6) +
2 3 2 4
1 1 1 1 5 7) + + 8) +
2 4 8 4 6
5 7 3𝑥 𝑥 9) + 10) +
6 12 4 12
8𝑦 4𝑦 5 −
7 11) + 12)
3 7 6𝑧 4𝑧 2 4
−3
−5
13) 14) 𝑎 𝑏 𝑎2 𝑎
6 7 1 −
5 15) 16)
𝑥 3𝑥 3𝑏2 + 6𝑏
4 5 1 1 17) + 18) +
9𝑦 12𝑦 2 3
3 5 −
2−
1 −
3 19) 20)
𝑥2 𝑥 2𝑥 6𝑎 8𝑎
9 1 1 −
5 21) − 22)
3𝑐 9𝑐2 12𝑟 6𝑟
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11 5 𝑥 2𝑥−
3𝑥 23) + 24) +
8𝑠 6𝑠2 10 5 2
𝑝 2𝑝−
5𝑝 𝑞−
5𝑞−
𝑞 25) + 26)
10 15 3 3 12 6
7 1 8 1 −
3 27) 28) + −
5𝑡 𝑡2 𝑤 5𝑤 10𝑤
𝑣−
𝑣 5 3 29) −1 30) + − 3
10 5 12𝑑 4𝑑
7 2 3𝑦 2 −
3𝑦 −
1 31) + 32) +
10𝑥 15 5𝑥 4 5𝑦 2
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Section 9.1: The Pythagorean Theorem
“Euclid alone has looked on beauty bare.” “Geometry is the art of reasoning well from ill-drawn figures.”
Given right triangle ABC with leg AC (also called leg b), leg BC (also called leg a) And hypotenuse AB (also called side c), find the missing side in each problem.
A
b c
C B a
1) Leg AC =3, Leg BC =4, find hypotenuse AB
2) Leg AC =5, Leg BC =12, find hypotenuse AB
3) Leg a=8, leg b=15, find hypotenuse c
4) Leg a=7, leg b=24, find hypotenuse c
5) Leg a=24, hypotenuse c=26, find leg b
6) Leg a=30, hypotenuse c=34, find leg b
7) Leg BC=40, hypotenuse AB=41, find leg AC
8) Leg BC=48, hypotenuse AB=50, find leg AC
9) Hypotenuse c=37, leg b=35, find leg a
10) Hypotenuse c=61, leg b=60, find leg a
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11) Side AC=6, side BC=8, find hypotenuse AB
12) Side AC=9, side BC=12, find hypotenuse AB
13) Side a=1, side b=1, find hypotenuse c
14) Side a=3, side b=3, find hypotenuse c
15) Side AC=2, side BC=3, find hypotenuse AB
16) Side AC=4, side BC=5 find hypotenuse AB
17) Hypotenuse AB=6, leg AC=4, find leg BC
18) Hypotenuse AB=6, leg AC=5, find leg BC
19) Hypotenuse c=6, leg a=4, find leg b
20) Hypotenuse c=6, leg a=3, find leg b
21) Side b=4, side a=8, find hypotenuse c
22) Side a=2, side b=6, find hypotenuse c
23) Side a=2, hypotenuse c=6, find side b
24) Side a=5, hypotenuse c=10, find side b
25) Side AC=5, side BC=10, find hypotenuse AB
26) Side AC= 3, side BC=6, find hypotenuse AB
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Section 9.2: Simplifying Radicals--Numerical; Variable
“Whether you think you can or you think you can’t—you’re right.”
Simplify:
1) √25 2) √49 3) −√144 4) −√81 5) √12 6) √20 7) −√32 8) −√50 9) 2√48 10) 4√27 11) −6√8 12) −10√24 13) 5√80 14) −15√50 15) √𝑥6 16) √𝑦8
17) −√𝑧10 18) −√𝑎12
19) √𝑏7 20) √𝑐13
21) √𝑑9 22) √𝑓25
23) −√𝑟16 24) √𝑠36
25) 2𝑥√𝑥3 26) 3𝑦√𝑦5
27) −4𝑧√𝑧7 28) −9𝑎2√𝑎11
29) √𝑥2𝑦4 30) √𝑎3𝑏4
31) −√𝑐10𝑑12 32) √𝑟2𝑠15
33) √12𝑥2 34) √75𝑦3
35) √27𝑐4 36) √72𝑟5
37) √8𝑥2𝑦7 38) √50𝑎4𝑏8
39) √54𝑐3𝑑11 40) √200𝑟9𝑠19
41) 4√32𝑥2𝑦3𝑧4 42) 6√72𝑎4𝑏5𝑐6
43) −10√18𝑟16𝑠9𝑡25 44) −14𝑥√27𝑥6𝑦10𝑧12
45) −2𝑎√144𝑎14𝑏20𝑐40 46) −5√80𝑢7𝑣3𝑤13
47) 7𝑥𝑦√20𝑥2𝑦5𝑧8 48) 12𝑎2𝑏𝑐3√50𝑎2𝑏7𝑐4
49) −24𝑟𝑠2𝑡5√18𝑟2𝑠10𝑡12 50) −20𝑢5𝑣9𝑤√72𝑢7𝑣3𝑤5
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Section 9.3 Multiplying Radicals
“You have a greater claim to the throne if you pull the sword from the stone
than if you say ‘I’m the king’ ”.
Multiply:
1) √3 × √5 3) √6 ∙ √30 5) √5 × √40 7) 2√3 ∙ 4√5 9) 6√5 × 3√15 11) 2𝑥√3𝑥3 ∙ 4𝑥√6𝑥2
13) 5𝑧2√2𝑧 ∙ 6𝑧4√8𝑧 15) −4𝑏√3𝑏 ∙ 8𝑏√3𝑏
17) 2𝑥√5𝑥𝑦𝑧 ∙ 6𝑥√10𝑥2𝑦3𝑧5
19) −8𝑥√2𝑥𝑦𝑧 ∙ 4𝑥√6𝑥2𝑦𝑧3
21) 3(√7 − 8) 23) √5(√5 + 2) 25) √5(√15 − √5) 27) 3√2(4√2 − 6√12)
29) (√3 + 5)(√6 + 4) 31) (√7 + 5)(√7 − 5) 33) (10 + √6)(10 − √6) 35) (√5 + 4)2
37) (√7 − 5)2
39) (√2 + √3)2
2) √2 × √7 4) √2 ∙ √8 6) √3 × √30 8) 6√7 ∙ 8√11 10) −2√3 × 4√27 12) −3𝑦√6𝑦7 ∙ 2𝑦√5𝑦4
14) 4𝑎√10𝑎8 ∙ 3𝑎√2𝑎5
16) 5𝑥𝑦√3𝑥2𝑦 ∙ 4𝑥𝑦√6𝑥𝑦
18) −2𝑎𝑏2√8𝑎𝑏5𝑐 ∙ 4𝑎2𝑏√6𝑎𝑏𝑐2
20) −7𝑟𝑠𝑡2√6𝑟3𝑠𝑡5 ∙ 10√3𝑟𝑠𝑡5
22) 4(√6 − 5) 24) √3(√3 + 6) 26) √2(5√2 − 4√8) 28) 5√3(16√6 − 3√27)
30) (√2 + 3)(√6 − 8) 32) (√3 + 6)(√3 − 6) 34) (12 − √8)(12 + √8) 36) (√6 + 3)2
38) (√8 − 9)2
40) (√5 + √6)2
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Section 9.4: Dividing Radicals
“I wish I didn’t know now what I didn’t know then.”
Divide:
1) √25
2) √49
36 81
√200 √48 3) 4)
√2 √3
√40 √200 5) 6)
√5 √10
14√15 9√24 7) 8)
2√5 3√12
6√20 24√12 9) 10)
3√2 6√3
√18𝑥3 √24𝑟3𝑠6 11) 12)
√2𝑥 √2𝑟𝑠2
√32𝑎5𝑏7 √40𝑥3𝑦4 13) 14)
√8𝑎3𝑏5 √10𝑥𝑦2
√5 √8 15) 16)
√3 √7
√12 √9 17) 18)
√50 √6
19) √7
20) √10
3 7
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21) √ 𝑥
24 22) √
𝑦2
5
23 √𝑥8
𝑦6 24) √20𝑧5
25
5 25)
√5
7 26)
√7
4√8∙√12 27)
2√3
2√6∙15√12 28)
5√2
8√15∙2√5 29)
4√3
7√3∙4√20 30)
15√5
4𝑥5√2𝑥7∙6𝑥3√45𝑥3 31)
8𝑥2√5𝑥
2𝑦3√3𝑦5∙4𝑦2√27𝑦7 32)
8𝑦√3𝑦
4√8∙5√6 33)
2√2
27√5∙3√10 34)
9√2
24√𝑥9 35)
√6𝑥
50√𝑦21 36)
√5𝑦
10 37)
√2 38) √
10
2
4 39)
√5+1
6 40)
√2−1
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Section 9.5 Combining Radicals
“Experience is what you get when you don’t get what you want.”
Combine:
1) 2√7 + 3√7
3) 8√5 + 4√5 − 6√5
5) 10√3𝑥 − 7√3𝑥 − 5√3𝑥
7) 7√5𝑎𝑏𝑐 − 4√5𝑎𝑏𝑐 − 6√5𝑎𝑏𝑐
9) 4√3 − 2√27
11) 5√12 − 6√75
13) 9√20 − 2√45
15) 6√32 + 8√200 − 5√98
17) 2√80 + 11√125 − 5√45
19) 8√20𝑥3 − 6𝑥√5𝑥
21) 5√𝑎2𝑏 + 6√4𝑎2𝑏 − 5𝑎√16𝑏
23) 2√12𝑥𝑦 + 4√27𝑥𝑦 − 5√75𝑥𝑦
25) 5√8 + 3√50 − 9√125
2) 11√3 − 5√3
4) 12√2 − 9√2 − 10√2
6) 14√6𝑥 + 2√6𝑥 − 3√6𝑥
8) 17√7𝑟𝑠𝑡 + 4√7𝑟𝑠𝑡 − 8√7𝑟𝑠𝑡
10) 7√2 + 4√8
12) 8√72 − 10√18
14) 3√50 − 5√8
16) 5√48 − 9√75 − 3√300
18) 3√8 + 6√72 − 9√50
20) 6√12𝑥5 − 8𝑥2√3𝑥
22) 7√𝑟𝑠3 + 9𝑠√𝑟𝑠 − 2𝑠√16𝑟𝑠
24) 3√5𝑦𝑧 − 4√45𝑦𝑧 − 9√20𝑦𝑧
26) 2√18 + 4√27 − 6√12
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Section 10.1: The Rectangular Coordinate System
Define the following terms 1) Horizontal Axis (X-Axis) 2) Vertical Axis (Y-Axis) 3) Origin 4) Quadrant I, Quadrant II, Quadrant III, Quadrant IV 5) Ordered Pair 6) Abscissa, Ordinate
Plot (Graph) the following points (Ordered Pairs) 7) A(2,4); B(1,6) 8) C(6,-1); D(8,-3) 9) E(-2,5); F(-5,3) 10) G(-1,-6); H(-3,-7) 11) K(0,4); L(0,-3) 12) M(1,0); N(-5,0)
13) P(4 ,1); Q(2, -2
1)
2 2
14) R(-1 1, 7); S(3
2 , 6)
3 3
Plot the following points on one coordinate axis 15) A(0,1); B(1,3); C(2,5); D(3,7); E(-1,-1)
Plot the following points on one coordinate axis 16) R(0,-1); S(1,2); T(2,5); U(3,8); V(-1,-4)
Find the coordinates for each of the following points (ordered pairs) 17) A; B 18) C; D 19) E; F 20) G; H 21) J; K
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-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
G
F
J
D
K
H
A
BC
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Section 10.2: Graphing By Use of the Table Method
Complete the given table and sketch the graph for:
1) 𝑦 = 𝑥 + 3 𝑥 𝑦 0 2
4
2) 𝑦 = 𝑥 − 2
3) 𝑦 = 3𝑥 + 2
4) 𝑦 = 4𝑥 − 3
5) 𝑥 + 𝑦 = 6
𝑥 𝑦 0 2 4
𝑥 𝑦 0 1 2
𝑥 𝑦 0 1 2
𝑥 𝑦 0 2 4
𝑥 𝑦 60
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6) 𝑥 − 𝑦 = 4
7) 2𝑥 + 3𝑦 = 12
8) 3𝑥 − 4𝑦 = 12
9) 6𝑥 + 3𝑦 − 18 =0
10) 2𝑥 − 5𝑦 − 10 = 0
11) 𝑦 = 1
𝑥 + 3 4
1 12) 𝑦 =
6𝑥 − 5
0 2
4
𝑥 𝑦 0 3 6
𝑥 𝑦 0 4 6
𝑥 𝑦 -2 0 3
𝑥 𝑦 -5 0
5
𝑥 𝑦 0 4
8
𝑥 𝑦 0 6
12
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Section 10.3: Graphing by Use of the Intercepts Method
Identify a) the Y-intercept and b) the X-intercept in each of the following equations. 1) 2𝑥 + 3𝑦 = 6 2) 3𝑥 + 4𝑦 = 12 3) 5𝑥 − 6𝑦 = 30 4) 7𝑥 − 3𝑦 = 21 5) 3𝑥 + 2𝑦 = 7 6) 4𝑥 + 6𝑦 = 9 7) 5𝑥 − 3𝑦 + 11 = 0 8) 8𝑥 − 5𝑦 − 12 = 0 9) 𝑦 = 2𝑥 − 3 10) 𝑦 = 5𝑥 − 8
Complete the given table and sketch the graph for each of the following equations.
11) 𝑥 + 𝑦 = 5 19) 2𝑥 − 3𝑦 = 9 𝑥 𝑦
0 0
2
𝑥 𝑦 0
0
2
12) 𝑥 − 𝑦 = 7 20) 3𝑥 − 4𝑦 = 8
𝑥 𝑦
0 0
4
𝑥 𝑦 0
0
2
13) 2𝑥 + 5𝑦 = 10
𝑥 𝑦
0 0
3
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14) 9𝑥 + 2𝑦 = 18 21) 𝑦 = 2𝑥 + 3
𝑥 𝑦
0 0
1
𝑥 𝑦 0
0
-1
15) 3𝑥 − 4𝑦 = 12 22) 3𝑥 − 4𝑦 = 8
𝑥 𝑦
0 0
2
𝑥 𝑦 0
0
2
16) 3𝑥 − 5𝑦 = 15
17) 𝑥 + 2𝑦 = 7
𝑥 𝑦
0 0
3
𝑥 𝑦
0 0
5
18) 2𝑥 + 3𝑦 = 11
𝑥 𝑦
0 0
3
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8
6
4
2
0
-8 -6 -4 -2 -2
0 2 4 6 8
-4
-6
-8
-10
Section 10.4: Graphing by Use of the Slope-Y Intercept Method
Find the slope, y-intercept, and sketch the graph for each of the following equations. 1) 𝑦 = 2𝑥 + 3 2) 𝑦 = 4𝑥 + 6 3) 𝑦 = 5𝑥 − 2 4) 𝑦 = 7𝑥 − 9
1 −1 5) 𝑦 =
2𝑥 + 3 6) 𝑦 =
4 𝑥 + 6
7) 𝑦 = 2𝑥 8) 𝑦 = 𝑥 9) 𝑦 = 4 10) 𝑦 = −2 11) 2𝑥 + 3𝑦 = 6 12) 4𝑥 + 5𝑦 = 20 13) 3𝑥 − 4𝑦 = 12 14) 5𝑥 − 2𝑦 = 10 15) 𝑥 + 4𝑦 = 24 16) 3𝑥 − 5𝑦 = 15
Write the equation of the line passing through the two given points in each of the following problems. 17) A(1,2); B(3,4) 18) C(2,1); D(-1,4) 19) E(3,5); F(6,8) 20) G(4,7); H(8,9) 21) K(-3,1); L(2,6) 22) M(-2,4); N(6,2) 23) P(-1,-3); Q(-2,-6) 24) R(-4,-7); S(-2,-5)
Write the equation of the vertical line passing through the given points. 25) S(2,5) 26) T(4,-6) 27) U(-8,9) 28) V(-7,-3)
Write the equation of the horizontal line passing through the given points. 29) S(2,5) 30) T(4,-6) 31) U(-8,9) 32) V(-7,-3)
Given the following graphs, write the equation of the line in slope and y-intercept form (𝑦 = 𝑚𝑥 + 𝑏) 33) 10
-10 10
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34)
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
-10 -8 -6 -4 -2 0 2 4 6 8 10
35)
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-8 -6 -4
8
6
4
2
0
-2 -2
-4
-6
-8
-10
0 2 4 6 8
-8 -6 -4
8
6
4
2
0
-2 -2
-4
-6
-8
-10
0 2 4 6 8
36) 10
-10 10
-10 10
37) 10
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Section 11.1: Solving a System of 2 Equations in 2 Unknowns Graphically
Solve the following systems of 2 equations in 2 unknowns graphically. If a system does not have a unique solution, identify the system as either dependent or inconsistent.
1) 𝑥 + 𝑦 = 4 2) 𝑥 + 𝑦 = 6 𝑥 − 𝑦 = 2 2𝑥 − 𝑦 = 6
3) 3𝑥 + 𝑦 = 6 4) 𝑥 − 𝑦 = 1 𝑥 + 𝑦 = 4 3𝑥 − 2𝑦 = 6
5) 𝑦 = 2𝑥 6) 𝑦 = 3𝑥 𝑦 = 𝑥 + 2 𝑦 = 2𝑥 + 1
7) 𝑥 + 𝑦 = 4 8) 𝑥 − 3𝑦 = −2 𝑦 = 2𝑥 − 5 𝑦 = 𝑥 + 2
9) 2𝑥 + 3𝑦 = 8 10) 3𝑥 + 4𝑦 = 10 4𝑥 + 𝑦 = 6 4𝑥 − 3𝑦 = 5
11) 2𝑥 + 3𝑦 = 6 12) 4𝑥 + 3𝑦 = 10 𝑦 = 4 𝑥 = 1
13) 3𝑥 + 5𝑦 = 15 14) 5𝑥 − 2𝑦 = 10 𝑥 + 2𝑦 = 5 2𝑥 − 𝑦 = 3
15) 5𝑥 − 3𝑦 = 15 16) 3𝑥 − 4𝑦 = 12 2𝑥 − 𝑦 = 7 𝑥 − 2𝑦 = 2
17) 𝑥 + 𝑦 = 2 18) 𝑥 − 𝑦 = 3 𝑥 + 𝑦 = 4 𝑦 = 𝑥 − 4
19) 𝑥 + 𝑦 = 3 20) 3𝑥 + 4𝑦 = 12 2𝑥 + 2𝑦 = 6 6𝑥 + 8𝑦 = 24
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Section 11.2: Solving a System of 2 Equations in 2 Unknowns by Use of the Substitution Method
Solve the following systems of equations by use of the Substitution Method.
1) 3𝑥 + 𝑦 = 15 2) 5𝑥 − 2𝑦 = 7 𝑦 = 3 𝑦 = 4
3) 2𝑥 − 3𝑦 = 4 4) 6𝑥 + 4𝑦 = 14 𝑥 = 5 𝑥 = 1
5) 𝑦 = 𝑥 + 2 6) 𝑦 = 𝑥 − 3 𝑥 + 𝑦 = 4 𝑥 + 2𝑦 = 6
7) 𝑥 = 𝑦 + 5 8) 𝑥 = 𝑦 − 1 2𝑥 − 𝑦 = 12 3𝑥 + 2𝑦 = 17
9) 𝑥 + 𝑦 = 6 10) 𝑥 + 𝑦 = 5 4𝑥 − 3𝑦 = 10 5𝑥 − 4𝑦 = −2
11) 2𝑥 + 𝑦 = 5 12) 3𝑥 + 𝑦 = 10 6𝑥 − 𝑦 = 3 7𝑥 − 3𝑦 = 2
13) 5𝑥 + 3𝑦 = 6 14) 3𝑥 − 2𝑦 = 2 4𝑥 + 2𝑦 = 6 2𝑥 + 4𝑦 = 28
15) 𝑥 − 𝑦 = −1 16) 2𝑥 − 𝑦 = 2 8𝑥 − 5𝑦 = 13 9𝑥 − 10𝑦 = 5
17) 3𝑎 − 2𝑏 = 4 18) 5𝑎 − 3𝑏 = 1 11𝑎 − 20𝑏 = 2 10𝑎 − 7𝑏 = −1
19) 4𝑟 + 𝑠 − 10 = 0 20) 2𝑟 − 𝑠 − 5 = 0 5𝑟 + 2𝑠 − 14 = 0 4𝑟 − 𝑠 − 1 = 0
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Section 11.3: Solving a System of 2 Equations in 2 Unknowns by Use of the Elimination Method (Adding or Subtracting)
Solve the following systems of equations by the Elimination Method.
1) 𝑥 + 𝑦 = 3 𝑥 − 𝑦 = 1
3) −𝑥 + 2𝑦 = 2 𝑥 + 3𝑦 = 13
5) 2𝑥 + 3𝑦 = 3 4𝑥 − 3𝑦 = 15
7) 𝑥 + 2𝑦 = 1 2𝑥 + 3𝑦 = −1
9) 4𝑥 + 3𝑦 = −2 3𝑥 + 𝑦 = −4
11) 5𝑥 + 4𝑦 = 13 6𝑥 − 3𝑦 = 39
13) 2𝑎 + 3𝑏 = 6 3𝑎 − 2𝑏 = −17
15) 7𝑟 − 4𝑠 = 22 5𝑟 + 6𝑠 = 60
17) 6𝑢 + 7𝑣 = 3 4𝑢 − 3𝑣 = 25
19) 𝑥 − 3𝑦 + 15 = 0
5𝑥 − 2𝑦 + 23 = 0
2) 2𝑥 + 𝑦 = 4 3𝑥 − 𝑦 = 1
4) −𝑥 + 𝑦 = 1 𝑥 + 2𝑦 = 11
6) 3𝑥 + 4𝑦 = 5 𝑥 − 4𝑦 = −9
8) 𝑥 − 3𝑦 = 7 3𝑥 + 2𝑦 = 10
10) 5𝑥 − 4𝑦 = −3 6𝑥 − 2𝑦 = −12
12) 6𝑥 + 3𝑦 = 6 9𝑥 − 5𝑦 = −48
14) 3𝑎 − 4𝑏 = 23 2𝑎 + 5𝑏 = 0
16) 8𝑟 + 3𝑠 = −2 3𝑟 − 4𝑠 = −11
18) 2𝑤 − 5𝑧 = 16 3𝑤 + 4𝑧 = 7
𝑥 −
𝑦 20) =1
2 6 𝑥
− 𝑦
= 5 3 4
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Section 12.1: Ratio and Proportion
“I was good at math before they decided to mix the alphabet into it.”
1) A car’s gas tank holds 20 gallons of fuel. If it can travel 500 miles on one tank full of gas, how many gallons of gas will it need to travel 120 miles?
2) A car uses 4 gallons of gas to go 120 miles. At that rate of usage, how many
gallons will it take to travel: a) 480 miles; b) 360 miles; c) 200 miles?
3) If 64 cups of coffee can be made using 1 pound of coffee, about how many
pounds of coffee will be needed to make 200 cups?
4) If a town in New York State assesses a real estate tax of $2.50 per $100 of
assessed valuation, what will the real estate tax be on a house with assessed
valuation of $11,000?
5) If 4 ounces of oil is needed to be added to 1 quart of gasoline to operate a chain
saw, how much oil will be needed for 1 gallon of gasoline (4 quarts)?
6) If a six foot tall man cast a four foot long shadow, how long will the shadow be
for a person of height of five feet?
7) If 6 ears of corn sells for $2.00, what will the cost be for a) 18 ears; b) 27 ears; c)
36 ears?
8) If a machine can produce 400 items in 8 hours, how long will it take to make: a)
1000 items; b) 1600 items; c) 2600 items?
9) If on average, 6 students receive A’s in a class of 30 students, how many calculus
students are expected to get A’s in a semester if there are: a) 450 students; b) 750
students; c) 1000 students?
10) If an instructor can grade 30 exam papers in one hour, how long will it take her
to grade: a) 100 papers; b) 150 papers; c) 175 papers?
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Section 12.2: Percent Problems
“I meant what I said and I said what I meant…An Elephant’s faithful one hundred percent!”
1) What is 6% of 200?
2) What is 9% of 400?
3) Five percent of what number is 300?
4) Eight percent of what number is 500?
5) Ninety percent of the students in a statistics class of 30 students passed their
first exam, if so, how many students actually passed this examination?
6) If 20% of a calculus class of 40 students received A’s as a final grade, how many
students received A’s?
7) Sixty percent of eligible voters voted in the most recent election. If there were
120 million eligible voters, how many voted?
8) A car can travel 500 miles on a full tank of gas. If it has travelled 350 miles,
a) what percent of its fuel has been used and b) what percent remains?
9) A new laptop computer has a list price of $800. If the price were reduced by
20%, a) what is the amount of the reduction and b) what is the new price?
10) A new car has a Manufacturer’s Suggested Retail Price (MSRP) of $25,000. If the cat can be purchased for 12% less than the MSRP, what is the new selling price?
11) A restaurant bill comes to $45.00. If one wants to leave a 20% tip, a) what is the
amount of the tip and b) what is the total cost of the meal?
12) An employee starts working at a new job at the rate of $12.00 per hour. If after 3
months, his salary is increase by 10%, what will his new salary be?
13) A light bulb manufacturing company found that 40 out of 1000 bulbs that it
produced were defective. What was the percent of defective bulbs produced?
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14) Thirty years ago, the cost to cross the Henry Hudson Bridge was 10 cents. Today
the cost is $2.50. A) What is the percent increase over the past 30 years? B) At this
rate, what will the cost to cross the bridge be 30 years from now?
15) A new car cost $20,000. If it depreciates (reduces in value) by 9% once you
drive it out of the showroom, how much will it be worth?
16) In a class of 25 students, 6 received B’s. What percent of the class received
grades of B?
17) New York State has a 7% sales tax on many items. If a television costs $500, a)
what is the tax on the TV and b) what is the total cost?
18) If you were to borrow $2,000 for one year at the interest rate of 5% per year
(compounded annually), a) how much interest would you owe and b) what amount
would you need to repay after one year?
19) At one college 40% of the entering class takes pre-calculus. If there are 2000
students in the entering class, a) how many take pre-calculus and b) how many
students do not take pre-calculus?
20) If a real estate broker receives a 5% commission on the sale of a house and she
receives a $20,000 commission, what is the selling price of the home?
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Section 12.3: Number Problems
“Smart people learn from their mistakes. Very smart people learn from other peoples’ mistakes.”
1) The sum of a given number and 7 is 25, what is the number?
2) The sum of a given number and 14 is 35, what is the number?
3) The sum of twice a number and 6 is 18, what is the number?
4) The sum of two consecutive numbers is 25, what are the two numbers?
5) The sum of two consecutive numbers is three times the original number, what is
the original number?
6) The sum of two numbers is 36. If one is twice the other, what are the two
numbers?
7) If three more than four times a number is 39. Find the number.
8) If four less than twice a number is 12, find the number.
9) If eight less than three times a number is 22, find the number.
10) If two times a number is increased by four, the result is 20. Find the original
number.
11) If a number is subtracted from 30, the result is 10. Find the number.
12) Six times a number is reduced by 5, the result is 67. Find the number.
13) If three times a number is increased by ten, the result is three less than four
times the original number. Find the original number.
14) Four times a number is decreased by five and equals seven more than twice the
original number. Find the original number.
15) A fifteen foot long board is cut into two pieces. Find the length of each piece if:
a) one is twice the length of the other; b) one is three feet longer than the other; and
c) one is three feet longer than twice the length of the other.
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Section 12.4: Distance = Rate × Time
“Happiness is not about what the world gives you—happiness is what you think
about what the world gives you.”
1) Two cars start from the same point and travel in opposite directions. If they each
travel at 60 miles per hour, how far apart will they be after 4 hours of travel?
2) Two small airplanes start at the same point and travel in exactly opposite
directions. If one travels at 200 miles per hour and the other travels at 150 miles per
hour, how far apart will they be after 3 hours of travel?
3) A man on a bicycle travels 48 miles in 6 hours. What was the average speed for
this trip?
4) If an airplane makes a 2500 mile trip in 5 hours, what was the average rate of
speed for the plane?
5) If an airplane travels 400 miles per hour for 3 ½ hours, how far did the plane go?
6) If it takes a jogger 12 minutes to run a mile, at that rate of speed, how many
miles will she cover in 2 hours?
7) A hiker can walk 32 miles in an 8 hour day. What is her average rate of speed for
this hike?
8) If a hiker can travel 12 miles per day over flat terrain, 8 miles per day over steep
terrain, and 24 miles per day in a canoe, how many miles will the hiker have
travelled in 6 days if there were 1 day of canoeing, 2 days of steep hiking, and 3 days
of flat hiking?
9) A truck driver leaves a factory at 8:00 am and travels till noon and then stops for
1 hour for a lunch break. If the truck driver then drives till 4:00 pm and averaged 55
miles per hour while driving, how far did he go?
10) The Pacific Coast Trail (PCT) is 2659 miles long. If a hiker covers the trail in 5
months (150 days), a) how many miles per day did she cover and b) if she hikes for
10 hours a day, what was her average rate per hour?
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11) The Pacific Coast Trail (PCT) is 4279 kilometers long. If a hiker covers the trail
in 5 months (150 days), a) how many kilometers per day did he cover and b) if he
hikes for 10 hours a day, what was her average rate per hour?
12) The Appalachian Trail (AT) is 2181 miles long. If a hiker covers the trail in 4
months (120 days), a) how many miles per day did she cover and b) if she hikes for
10 hours a day, what was her average rate per hour?
13) The Appalachian (AT) is 3510 kilometers long. If a hiker covers the trail in 4
months (120 days), a) how many miles per day did he cover and b) if he hikes for 10
hours a day, what was her average rate per hour?
14) Two cars start from the same point and travel in exactly opposite directions. If one travels at 65 miles per hour and the other at 50 miles per hour, after 6 hours: a) how far apart will the cars be; b) if they each can go 30 miles on 1 gallon of gasoline, how much gas will be used ; and c) if gas costs $2.50 per gallon, how much must each car pay?
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Answers to Odd Numbered Problems
Section 1.1: Nomenclature-Terms, Constants, Monomials, Binomials, Trinomials, Absolute Values
1) Coefficent: 8; Variable: x 3) Coefficent: 7; Variables: x and y 5) Coefficent: 12; Variables: x, y and z 7) Monomial 9) Monomial 11) Trinomial 13) Terms: 4𝑥2, 8𝑥, −10 15) Terms: −6𝑥3, 7𝑥2, −3𝑥, 4 17) Variables: 10𝑥2, −7𝑥 ; Constant: 2 19) Variables: −7𝑥4, 3𝑥2, −4𝑥 ; Constant: 17 21) Degree: 2; Number of Terms: 3; Constant: -7 23) Degree: 16; Number of Terms: 4; Constant: -21 25) Degree: 7; Number of Terms: 5; Constant: none 27) 4 29) 24
3 31)
8 33) 11
Section 1.2: Exponents 1) 36 = 729 3) 16 5) -16 7) 𝑥8
9) 𝑚11
11) 𝑎10𝑏15
13) −20𝑥7𝑦5
15) 21𝑑3𝑓5
17) 2𝑥6
19) 8𝑧16
21) 16𝑎8𝑏12
23) 𝑎2
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25) 𝑐4
1 27)
𝑝6
1 29)
𝑟2 1
31) 𝑡8
33 ) 𝑤13
1 35)
𝑦11
16 37)
9 9
39) 4 1
41) 16
43) −1
16 45) b
128 47)
𝑟18
49) 625𝑥8
125𝑥6
51) 𝑦12
53) −16𝑎5
Section 1.3: Scientific Notation 1) 2.4 X 103
3) 2.8 X 102
5) 3 X 10−2
7) 4.5 X 10−7
9) 480,000 11) 8,140,000,000 13) .094 15) .356 17) 6 X 107 = 60,000,000 19) 4.212 X 1017 = 421,200,000,000,000,000 21) 8 X 10−8 = .00000008 23) 2.34 X 10−7 = .000000234 25) 3 X 102 = 300
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27) 3 X 10−8 = .00000003 29) 2 31) 1.02 X 106 = 1,020,000 33) 1 X 10−6 = .000001
Section 2.1: Operations Involving Addition, Subtraction, Multiplication, or Division 1) 30 3) -21 5) -10 7) -45 9) 21 11) 33 13) -36 15) -160 17) 6 19) -8 21) -2 23) 2 25) 9 27) -2 29) 0 31) 10 33) 5 35) 7 37) -21 39) 12 41) 20 43) -7 45) -5 47) 18 49) -5 51) 0 53) 0 55) 22 57) 32 59) -3 61) 4
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Section 2.2: Order of Operations 1) 34 3) 0 5) 5 7) 20 9) 40 11) 6 13) 19 15) 28 17) 8 19) 17 21) 2 23) 1 25) 2 27) -5 29) -1 31) -17 33) 79 35) 26 37) 10 39) -96
Section 3.1: Evaluating Algebraic Expressions 1) 12 3) 18 5) 48 7) 36 9) 1 11) 8 13) -9 15) -28 17) 9 19) 12 21) -8 23) 4
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25) -4 27) 12
Section 3.2: Functional Notation 1) 17 3) 1 5) -12 7) -39 9) -34 11) 4 13) 9 15) 6 17) 3 19) 0 21) 14 23) 2 25) 4a+4h+3 27) 𝑎2 + 2𝑎ℎ + ℎ2 + 4𝑎 + 4ℎ + 6 29) −4𝑎2 + 2𝑎 − 6
Section 4.1: Multiply Monomial ∙ Monomial
1) 𝑥5
3) 20𝑎8
5) −18𝑠12
7) 24𝑎3𝑏3
9) −10𝑟6𝑠4
11) 9𝑥8𝑦10
13) 64𝑥4
15) 54𝑥4𝑦4
17) 90𝑟7𝑠12
19) 60𝑥3𝑦5𝑧9
21) 24𝑎5𝑏14
23) −24𝑥5𝑦9𝑧8
25) −30𝑝11𝑞12𝑟15
27) −18𝑎5𝑏8
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29) 180𝑥7𝑦4
31) −20𝑥6
Section 4.2: Multiply Monomial ∙ Binomial—The Distributive Law 1) 16𝑥 + 48 3) 18𝑥3 + 42𝑥 5) −2𝑦4 − 6𝑦2
7) −27𝑧4 + 45𝑧3
9) 2𝑎4 + 5𝑎3
11) 44𝑏2 − 8𝑏3
13) −2𝑟3 + 20𝑟2 − 18𝑟 15) 3𝑥4𝑦2 − 4𝑥3𝑦3 + 9𝑥2𝑦 17) −2𝑥4 − 10𝑥3 + 12𝑥2
19) −20𝑟7𝑠5 − 24𝑟6𝑠4 − 16𝑟6𝑠3 + 4𝑟4𝑠3
Section 4.3: Binomial ∙ Binomial 1) 𝑥2 + 4𝑥 + 3 3) 𝑥2 + 2𝑥 − 35 5) 𝑦2 + 10𝑦 + 16 7) −𝑦2 + 𝑦 + 6 9) 2𝑎2 + 21𝑎 + 27 11) 12𝑏2 + 11𝑏 − 15 13) 12𝑐2 − 28𝑐 + 15 15) 15𝑥2 − 25𝑥 + 10 17) 18𝑥4 − 69𝑥3 − 12𝑥2
19) 𝑥2 − 6𝑥 + 9 21) 9𝑥2 − 42𝑥 + 49 23) 𝑥2 − 16 25) 36−𝑦2
27) 16𝑥2 − 25 29) 20𝑥3 + 20𝑥2 + 5𝑥
Section 4.4: Multiply Binomial ∙ Trinomial or Trinomial ∙ Trinomial 1) 𝑥3 + 3𝑥2 + 5𝑥 + 3 3) 𝑥3 − 5𝑥2 + 14𝑥 − 24
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5) 2𝑦3 − 3𝑦2 − 32𝑦 + 48 7) 5𝑧3 − 5𝑧2 + 34𝑧 − 48 9) 4𝑧4 + 36𝑧3 + 6𝑧2 + 51𝑧-27 11) 𝑎3 + 9𝑎2 + 15𝑎 + 7 13) 𝑏3 + 3𝑏2 + 3𝑏 + 1 15) 8𝑟3 + 12𝑟2 + 6𝑟 + 1 17) 𝑥4 + 5𝑥3 + 8𝑥2 + 5𝑥 + 7 19) 2𝑥4 + 18𝑥3 + 16𝑥2 − 120𝑥
Section 5.1: Combining Like Terms 1) −11𝑥 3) 8𝑥2𝑦 5) 2𝑥 + 16𝑦 7) 23𝑝 − 32 9) 7𝑟𝑠2 + 10 11) 2𝑥2𝑦 + 10𝑥𝑦2 − 8𝑥𝑦 13) −11𝑎3 − 24𝑎𝑏 − 7𝑎 15) 4𝑥2 + 𝑥 − 5 17) 3𝑥3 − 15𝑥2 + 5𝑥 − 16 19) 3𝑥2 + 19𝑥 − 27 21) −6𝑐3𝑑 + 9𝑐2𝑑2 − 12𝑐2𝑑3 + 5𝑐2𝑑 23) 9 − 14𝑥 + 14𝑦 25) 4𝑥2 − 5𝑥 − 2 27) −34𝑟3𝑠3 + 18𝑟2𝑠2 + 10𝑟𝑠 29) 16𝑝2𝑞 − 9𝑝2𝑞2
31) 10𝑥3𝑦2 − 26𝑥 − 2𝑥3𝑦 + 2𝑥2
33) 0 35) −38𝑥2 + 49𝑥 − 45
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Section 6.1: Unique Prime Factorization of Integers 1) 22 ∙ 3 3) 25 ∙ 3 5) 2 ∙ 32
7) 22 ∙ 5 9) 23 ∙ 32
11) 32 ∙ 11 13) 24 ∙ 3 15) 24 ∙ 5 17) 22 ∙ 33
19) 2 ∙ 3 ∙ 7 21) 33
23) 25 ∙ 3 ∙ 5 25) 2 ∙ 53
27) 24 ∙ 32 ∙ 5 29) 23 ∙ 32 ∙ 5
Section 6.2: Greatest Common Monomial Factor (GCF)-Numerical or Algebraic 1) 2(𝑥 − 2) 3)𝑧(5𝑧 + 8) 5) 4𝑦(𝑦2 + 5𝑦 − 2) 7) 7𝑏(𝑏2 − 2𝑏 + 3) 9) 12𝑥𝑦3(3𝑥 + 1) 11) 7𝑟(2𝑟2 − 4𝑟 + 5) 13) 5𝑎3(4𝑎7 + 2𝑎2 − 1) 15) 9𝑥2𝑦2𝑧2(2𝑥𝑦2𝑧3 − 3𝑦𝑧 + 1) 17) 11𝑎2𝑏𝑐(1 − 2𝑏 + 3𝑏𝑐) 19) 𝑦𝑧(2𝑥𝑧 − 3𝑥 + 4) 21) 4(𝑥3 + 2) 23) 15𝑥𝑦2(1 + 2𝑦) 25) 2(3𝑥2 + 8𝑦2 + 13𝑧2) 27) 4𝑥𝑦(2𝑥2𝑦3 + 4𝑥𝑦2 + 1) 29) 10𝛼(4 + 2𝛽 + 𝜇)
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Section 6.3: The Difference of Two Squares (DOTS) 1) (𝑥 + 3)(𝑥 − 3) 3) (𝑦 + 8)(𝑦 − 8) 5) (10 + 𝑧)(10 − 𝑧) 7) Not Factorable 9) Not Factorable 11) (𝑥2 + 1)(𝑥 + 1)(𝑥 − 1) 13) (𝑦4 + 12)(𝑦4 − 12) 15) (6𝑦8 + 11)(6𝑦8 − 11) 17) (4𝑦8 + 7)(4𝑦8 − 7) 19) (𝑔3ℎ2 + 𝑚)(𝑔3ℎ2 − 𝑚) 21) (3 + 5𝑦)(3 − 5𝑦) 23) (9𝑥 + 2𝑧)(9𝑥 − 2𝑧) 25) (𝑥2 + 𝑦)(𝑥2 − 𝑦) 27) (5𝑎3𝑏4 + 2)(5𝑎3𝑏4 − 2) 29) (𝑔3ℎ2 + 𝑚)(𝑔3ℎ2 − 𝑚) 31) (𝑎8𝑏18 + 6)(𝑎8𝑏18 − 6) 33) (𝑚2 + 𝑛2)(𝑚 + 𝑛)(𝑚 − 𝑛)
Section 6.4: Combination of Greatest Common Factor and Difference of Two Squares
1) 2(𝑥2 − 1) = 2(𝑥 + 1)(𝑥 − 1) 3) 6𝑧(1 − 4𝑧2) = 6𝑧(1 + 2𝑧)(1 − 2𝑧) 5) 2(1 − 4𝑏2) = 2(1 + 2𝑏)(1 − 2𝑏) 7) 4(𝑑2 − 25) = 4(𝑑 + 5)(𝑑 − 5) 9) 2(𝑦4 − 16) = 2(𝑦2 + 4)(𝑦2 − 4) = 2(𝑦2 + 4)(𝑦 + 2)(𝑦 − 2) 11) 8(9𝑎2 − 1) = 8(3𝑎 + 1)(3𝑎 − 1) 13) 16(1 − 𝑐16) = 16(1 + 𝑐8)(1 − 𝑐8) = 16(1 + 𝑐8)(1 + 𝑐4)(1 − 𝑐4)
= 16(1 + 𝑐8)(1 + 𝑐4)(1 + 𝑐2)(1 − 𝑐2) = 16(1 + 𝑐8)(1 + 𝑐4)(1 + 𝑐2)(1 + 𝑐)(1 − 𝑐)
15) 𝑥2(𝑥2 − 121) = 𝑥2(𝑥 + 11)(𝑥 − 11) 17) 𝑎2(4𝑏2 − 𝑎2) = 𝑎2(2𝑏 + 𝑎)(2𝑏 − 𝑎) 19) 2𝑥2𝑦3𝑧2(9𝑧2 − 1) = 2𝑥2𝑦3𝑧2(3𝑧 + 1)(3𝑧 − 1) 21) 𝑎3(𝑎2 − 1) = 𝑎3(𝑎 + 1)(𝑎 − 1) 23) 𝑥2(25𝑥2 − 4) = 𝑥2(5𝑥 + 2)(5𝑥 − 2) 25) 9(𝑥2 − 9) = 9(𝑥 + 3)(𝑥 − 3)
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27) 𝑎(𝑎2 − 144) = 𝑎(𝑎 + 12)(𝑎 − 12) 29) 9𝑐𝑑(𝑐2𝑑2 − 4) = 9𝑐𝑑(𝑐𝑑 + 2)(𝑐𝑑 − 2)
Section 6.5: Trinomial With Leading Coefficient One (𝑥2 + 𝑏𝑥 + 𝑐) 1) (𝑥 + 2)(𝑥 + 1) 3) (𝑥 + 2)(𝑥 − 1) 5) (𝑦 + 3)(𝑦 + 2) 7) (𝑦 − 3)(𝑦 + 2) 9) (𝑧 + 3)(𝑧 + 1) 11) (𝑧 − 3)(𝑧 + 1) 13) (𝑎 + 6)(𝑎 + 4) 15) (𝑐 − 4)(𝑐 − 2) 17) (𝑠 − 6)(𝑠 − 3) 19) (𝑤 − 5)(𝑤 + 3) 21) (𝑥 − 8)(𝑥 + 6) 23) (𝑧 − 15)(𝑧 − 2) 25) (𝑏 + 5)(𝑏 + 5) 27) (𝑑 − 14)(𝑑 − 2) 29) (𝑦 + 11)(𝑦 − 6) 31) (𝑢 + 5)(𝑢 + 1) 33) (𝑏 − 8)(𝑏 + 2) 35) (𝑑 + 5)(𝑑 + 4) 37) (𝑦 + 10)(𝑦 + 4) 39) (𝑎 + 8)(𝑎 + 6)
Section 6.6: Trinomial With Leading Coefficient Other Than One (𝑎𝑥2 + 𝑏𝑥 + 𝑐)
1) (2𝑎 + 1)(𝑎 + 2) 3) (2𝑥 + 1)(𝑥 − 2) 5) (3𝑦 − 1)(𝑦 − 2) 7) (3𝑦 − 2)(𝑦 − 1) 9) (5𝑧 + 1)(𝑧 − 3) 11) (5𝑧 − 1)(𝑧 − 3) 13) (3𝑎 + 2)(𝑎 − 1) 15) (3𝑐 − 4)(𝑐 − 1)
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17) 2(2𝑔2 + 5𝑔 − 12) = 2(2𝑔 − 3)(𝑔 + 4) 19) (3𝑠 − 5)(2𝑠 + 1) 21) (4𝑢 + 3)(2𝑢 + 3) 23) (3𝑤 + 4)(3𝑤 − 2) 25) (6𝑦 − 5)(2𝑦 + 3) 27) (5𝑎 − 3)(3𝑎 + 2) 29) (4𝑐 + 1)(4𝑐 + 1)
Section 6.7: Combination of Greatest Common Monomial Factor and Trinomial 1) 4(𝑥2 − 𝑥 − 2) = 4(𝑥 − 2)(𝑥 + 1) 3) 𝑧3(𝑧2 − 2𝑧 − 3) = 𝑧3(𝑧 − 3)(𝑧 + 1) 5) 6(𝑏2 − 𝑏 − 20) = 6(𝑏 − 5)(𝑏 + 4) 7) 4(𝑏2 + 5𝑏 + 6) = 4(𝑏 + 2)(𝑏 + 3) 9) 5𝑠2(𝑠2 − 6𝑠 − 7) = 5𝑠2(𝑠 − 7)(𝑠 + 1) 11) 4𝑢2(𝑢2 + 3𝑢 + 2) = 4𝑢2(𝑢 + 2)(𝑢 + 1) 13) 6(𝑥2 + 5𝑥 + 6) = 6(𝑥 + 3)(𝑥 + 2) 15) 8(𝑦2 + 4𝑦 + 3) = 8(𝑦 + 3)(𝑦 + 1) 17) 11(𝑎2 − 6𝑎 + 8) = 11(𝑎 − 4)(𝑎 − 2) 19) 15(𝑐2 − 3𝑐 − 10) = 15(𝑐 − 5)(𝑐 + 2)
Section 6.8: Factoring by Grouping 1) (𝑥 + 4)(𝑥 + 2) 3) (𝑧 + 9)(𝑧 − 7) 5) (𝑏 + 7)(4𝑏 + 3) 7) (𝑑 − 8)(7𝑑 − 5) 9) (𝑥 − 6)(𝑥2 + 4) 11) (𝑧 − 4)(𝑧2 + 3) 13) (𝑥 + 2)(2𝑥 + 3) 15) (4𝑧 − 1)(𝑧 − 2) 17) (𝑥 + 3𝑦)(3𝑥2 − 4) 19) (𝑏 + 3𝑐)(5𝑎2 − 6) 21) (7𝑣 + 2𝑤)(9𝑢2 + 5𝑥)
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Section 6.9: Review of all Factoring 1) (6 + 𝑥)(6 − 𝑥) 3) 7𝑎𝑏(4𝑎𝑏2 − 5𝑎𝑏 + 1) 5) (𝑥2 + 1)(𝑥2 − 1) = (𝑥2 + 1)(𝑥 + 1)(𝑥 − 1) 7) (4𝑎8 + 5)(4𝑎8 − 5) 9) 𝑦(4𝑥𝑧 + 5𝑥 + 6𝑧) 11) 𝑐2(𝑏4 − 1) = 𝑐2(𝑏2 + 1)(𝑏2 − 1) = 𝑐2(𝑏2 + 1)(𝑏 + 1)(𝑏 − 1) 13) (𝑟 + 10)(𝑟 − 3) 15) Not Factorable 17) 3𝑣2(5𝑣 + 4) 19) 4𝑥(𝑥 − 1) 21) 5𝑥𝑦(4𝑥 − 5𝑦 + 6) 23) (𝑥2 + 4)(𝑥2 − 4) = (𝑥2 + 4)(𝑥 + 2)(𝑥 − 2) 25) (2𝑧 + 3)(𝑧 − 5) 27) (7 + 𝑏)(3 + 𝑏) 29) (𝑥 + 5)(𝑥 − 5) 31) (6𝑥 − 5)(𝑥 + 1) 33) (2𝑥 + 3𝑦)(3𝑧2 + 5)
Section 7.1: Is a Given Number a Solution to an Equation 1) 5 = 5, Yes 3) −2 = −2, Yes 5) −3 ≠ 6, No 7) 8 ≠ 0, No 9) 4 ≠ 0, No 11) 12 ≠ 0, No 13) −16 ≠ 0, No 15) 12 ≠ −4, No 17) 37 ≠ 36, No
Section 7.2: Linear (First Degree) Equations 1) 𝑥 = 9 3) 𝑥 = −4 5) 𝑦 = −7 7) 𝑦 = 9 9) 𝑧 = 9
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11) 𝑎 = 16 13) 𝑏 = 7 15) 𝑥 = 0 17) 𝑐 = 3 19) 𝑑 = 6
21) 𝑟 = 3⁄2 23) 𝑠 = 7⁄2 25) 𝑡 = 6
27) 𝑣 = 14⁄5 29) 𝑥 = 1 31) 𝑧 = 2 33) 𝑧 + 10
Section 7.3: Decimal Equations 1) 𝑥 = 2 3) 𝑥 = 2 5) 𝑦 = 2 7) 𝑧 = 200 9) 𝑧 = −1
11) 𝑎 = 4⁄3 13) 𝑏 = 140⁄9 15) 𝑐 = 1 17) 𝑟 = −1 19) 𝑠 = 40 21) 𝑡 = −3
23) 𝑤 = 9⁄5 25) 𝑥 = 2 27) 𝑦 = 3000 29) 𝑧 = 2
Section 7.4: Fractional Equations with Monomials or Binomials in the Numerator 1) 𝑥 = 20 3) 𝑥 = −16 5) 𝑦 = 51
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7) 𝑧 = 4 9) 𝑥 = −3 11) 𝑏 = −5 13) 𝑠 = 3 15) 𝑥 = 12 17) 𝑧 = 4 19) 𝑥 = 8
21) 𝑥 = −3⁄5 23) 𝑥 = 5 25) 𝑥 = 9 27) 𝑥 = 0
29) 𝑥 = 155⁄29
Section 7.5: Literal Equations
1) 𝑟 = 𝑑⁄𝑡 3) 𝐼 = 𝑉⁄𝑅 5) ℎ = 2𝐴⁄𝑏 7) 𝑅 = 𝑃𝑉⁄𝑛𝑇 9) 𝑚 = 2𝑘⁄𝑣2
(𝑣2 − 𝑣1) 11) 𝑇 = ⁄𝑎 13) 𝑏 = 𝑃 − 𝑎 − 𝑐
(𝑆 − 2𝜋𝑟2) 15) 𝑙 = ⁄
2𝜋𝑟 (𝐹𝑟2)
17) 𝑚1 = ⁄𝐺𝑚2 (𝐼 − 𝑃)
19) 𝑟 = ⁄𝑃𝑡 𝑦2
21) 𝑥 = ⁄4𝑝
23) 𝑟 = 𝑚𝑣2⁄𝐹𝑔
𝑃1𝑉1𝑇2 25) 𝑣2 = ⁄𝑃2𝑇1 (4𝑥 − 20)
27) 𝑦 = ⁄5
(10 − 2𝑥 − 3𝑦) 29) 𝑧 = ⁄4
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Section 7.6: Quadratic (Second Degree) Equations 1) 𝑥 = 2, 5 3) 𝑦 = 0, 6 5) 𝑧 = 0, 9 7) 𝑎 = 8, −8 9) 𝑥 = 4, −3 11) 𝑥 = 10, −2
13) 𝑦 = −5, −1⁄2 15) 𝑦 = −2, 3⁄4 17) 𝑧 = 2, −1 19) 𝑧 = 3, −1, 0 21) 𝑎 = 6, −2 23) 𝑐 = 2, −1 25) 𝑟 = 4, 2 27) 𝑡 = 4, 0 29) 𝑣 = 9, −9 31) 𝑥 = −1, −2 33) 𝑥 = 2, −3 35) 𝑦 = 2 37) 𝑥 = 1, −1 39) 𝑧 = −2, −1
Section 7.7: Linear Inequalities 1) 𝑥 > 5 3) 𝑥 ≥ 4 5) 𝑥 < −5 7) 𝑥 ≤ −2 9) 𝑥 ≥ −2 11) 𝑥 < 2 13) 𝑥 > 15 15) 𝑥 ≤ −28 17) 𝑥 ≥ 12 19) 𝑥 < −28 21) 𝑥 < 2 23) 𝑥 ≤ −27
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25) 𝑥 ≥ 4 27) 𝑥 > 2 29) 𝑥 ≥ −4 31) 𝑥 < 1 33) 𝑥 < −15 35) 𝑥 ≤ 5 37) 𝑥 > 4 39) 𝑥 ≤ −1
Section 8.1: Simplifying Fractions (Rational Expressions) 1) 3 3) – 𝑥2
5) 2𝑥 −1
7) 3𝑝2𝑞
9) −6𝑎𝑏4𝑐5
11) 4𝑥2 − 2𝑥 + 1 13) 4𝑎2𝑏 − 2𝑎 + 1 15) −25𝑐19 − 16𝑐7 + 9𝑐3
17) -1 4(𝑥−1) 4
19) = (𝑥+1)(𝑥−1) (𝑥+1)
6𝑧(𝑧−2) 6𝑧 21) =
(𝑧+2)(𝑧−2) (𝑧+2) (𝑏−4)(𝑏+3) (𝑏+3)
23) = (𝑏+4)(𝑏−4) (𝑏+4)
3𝑟(𝑟−1) 3𝑟 25) =
(𝑟+6)(𝑟−1) (𝑟+6) (𝑡−4)(𝑡+3) (𝑡−4)
27) = (𝑡+3)(𝑡+2) (𝑡+2) (𝑧−4)(𝑧−2) (𝑧−2)
29) = 2(𝑧−4) 2
6𝑥2𝑦3
31) 𝑧
Section 8.2: Multiplying Fractions (Rational Expressions) 1) 1 3) 10
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5) 6𝑎
7) 𝑏2
9) 2𝑎𝑏 11) 14𝑥𝑧 13) 3
−4𝑥𝑦 15)
15 17) −4𝑟4𝑠2
19) −3𝑥2𝑦8𝑧5
1 21)
2
Section 8.3: Dividing Fractions (Rational Expressions) 3
1) 2
3) 6
𝑎2
5) 𝑏2 −8𝑦
7) 𝑏𝑥
9𝑑 9)
2 1
11) 9𝑏
13) 𝑎3𝑝3𝑞r −6𝑥𝑦
15) 𝑐
17) −16𝑟4𝑠2𝑝𝑞 19) 15𝑦2
8𝑦3
21) 𝑏 4
23) 3𝑎𝑞
𝑥 25)
𝑥−1 𝑦(𝑥−3)
27) 𝑥−4
Section 8.4: Adding or Subtracting Fractions (Rational Expressions)
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1) 1 1
3) 5 5
5) 6 7
7) 8 17
9) 12
8𝑦 11)
21 (2𝑏−3𝑎)
13) 𝑎𝑏
13 15)
3𝑥 31
17) 36𝑦 (6−3𝑥)
19) 2𝑥2
(27𝑐−1) 21)
9𝑐 (33𝑠+20)
23) 24𝑠2
−43𝑝 25)
30 (7𝑡−15)
27) 5𝑡2
(3𝑣−10) 29)
10 (4𝑥+15)
31) 30𝑥
Section 9.1: The Pythagorean Theorem 1) AB=5 3) c=17 5) b=10 7) AC=9 9) a=12 11) AB=10
13) c=√2
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15) AB=√13 17) BC=2√5 19) b=2√5 21) c=4√3 23) b=4√2 25) AB=5√5
Section 9.2: Simplifying Radicals 1) 5 3) −12 5) 2√3 7) −4√3 9) 8√3 11) −12√2 13) 20√5 15) 𝑥3
17) – 𝑧5
19) 𝑏3√𝑏 21) 𝑑4√𝑑 23) – 𝑟8
25) 2𝑥2√𝑥 27) −4𝑧4√𝑧 29) 𝑥𝑦2
31) – 𝑐5𝑑6
33) 2𝑥√3 35) 3𝑐2√3 37) 2𝑥𝑦3√2𝑦
39) 3𝑐𝑑5√6𝑐𝑑 41) 16𝑥𝑦𝑧2√2𝑦
43) −30𝑟8𝑠4𝑡12√2𝑠𝑡 45) −24𝑎8𝑏10𝑐20
47) 14𝑥2𝑦3𝑧4√5𝑦 49) −72𝑟2𝑠7𝑡11√2
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Section 9.3 Multiplying Radicals
1) √15 3) 6√5 5) 10√2 7) 8√15 9) 90√3 11) 24𝑥4√2𝑥 13) 120𝑧7
15) −96𝑏3
17) 60𝑥3𝑦2𝑧3
19) −64𝑥3𝑦𝑧2√3𝑥 21) 3√7 − 24 23) 5 + 2√3 25) 5√3 − 5 27) 24 − 36√6 29) 3√2 + 4√3 + 5√6 + 20 31) −23 33) 94
35) 21 + 8√5 37) 32 − 10√7 39) 5 + 2√6
Section 9.4: Dividing Radicals 5
1) 6
3) 10
5) 2√2 7) 7√3 9) 2√10 11) 3𝑥 13) 2𝑎𝑏
√15 15)
3
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5
√6 17)
√21 19)
3 √6𝑥
21) 12
𝑥4
23) 𝑦3
25) √5 27) 8√2 29) 20
31) 9𝑥10√2𝑥 33) 20√6 35) 4𝑥4√6 37) 5√2 39) √5 − 1
Section 9.5: Combining Radicals
1) 5√7 3) 2√5 5) −2√3𝑥 7) −3√5𝑎𝑏𝑐 9) −2√3 11) −20√3 13) 12√5 15) 69√2 17) 48√5 19) 10𝑥√5𝑥 21) −3𝑎√𝑏 23) −9√3𝑥𝑦
25) 25√2 − 45√5
Section 10.1: The Rectangular Coordinate System
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1) One of two perpendicular lines intersecting at the origin. It is also called the abscissa and indicates how far left or right of the origin the point is. 3) The point where the x-axis and y-axis meet indicated by (0, 0) 5) Two numbers usually written with parenthesis (a, b) where the first number indicates how far to the left or right the point is and the second number indicates how high up or down. 17) A (2, 6) and B (8, 3) 19) No point E and F (2, 0) 21) J (4, -4) and K (-8, -3)
Section 10.2 Graphing by Use of the Table Method 1) 𝑦 = 𝑥 + 3
𝑥 𝑦 0 3 2 5
4 7
2) 𝑦 = 𝑥 − 2
3) 𝑦 = 3𝑥 + 2
4) 𝑦 = 4𝑥 − 3
𝑥 𝑦 0 -2 2 0 4 2
𝑥 𝑦 0 2 1 5 2 8
𝑥 𝑦 0 -3 1 1
2 5
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5) 𝑥 + 𝑦 = 6 𝑥 𝑦 0 6 2 4 4 2
6) 𝑥 − 𝑦 = 4
7) 2𝑥 + 3𝑦 = 12
8) 3𝑥 − 4𝑦 = 12
9) 6𝑥 + 3𝑦 − 18 =0
10) 2𝑥 − 5𝑦 − 10 = 0
𝑥 𝑦 0 -4 2 -2
4 0
𝑥 𝑦 0 4 3 2
6 0
𝑥 𝑦 0 -3 4 0 6 1.5
𝑥 𝑦 -2 10 0 6 3 0
𝑥 𝑦 -5 -4 0 -2 5 0
𝑥 𝑦
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1 11) 𝑦 =
4𝑥 + 3 0 3
4 4
8 5
1 12) 𝑦 =
6𝑥 − 5 𝑥 𝑦
0 -5 6 -4 12 -3
Section 10.3: Graphing by Use of the Intercepts Method 1) The x-intercept is (0, 2) and the y-intercept is (3, 0) 3) The x-intercept is (0, -5) and the y-intercept is (6, 0)
7 7 5) The x-intercept is (0, ) and the y-intercept is ( , 0)
2 3 −11 −11
7) The x-intercept is (0, ) and the y-intercept is ( , 0) 3 5
3 9) The x-intercept is (0, -3) and the y-intercept is ( , 0)
2
11) 𝑥 + 𝑦 = 5
𝑥 𝑦
0 5 5 0
2 3
13) 2𝑥 + 5𝑦 = 10
𝑥 𝑦
0 2 5 0
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3 4
5
15) 3𝑥 − 4𝑦 = 12
17) 𝑥 + 2𝑦 = 7
𝑥 𝑦
0 -3 4 0
2 −3
2
𝑥 𝑦
0 7
2 7 0
5 1
𝑥 𝑦
0 -3 9
2 0
2 −5
3
19) 2𝑥 − 3𝑦 = 9
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21) 𝑦 = 2𝑥 + 3
𝑥 𝑦
0 3 −3
2 0
-1 1
Section 10.4 Graphing by Use of the Slope-Y Intercept Method 1) Slope is m=2 and the y-intercept is (0, 3) 3) Slope is m=5 and the y-intercept is (0, -2)
1 5) Slope is m= and the y-intercept is (0, 3)
2 7) Slope is m=2 and the y-intercept is (0, 0) 9) Slope is m=0 and the y-intercept is (0, 4)
−2 11) Slope is m= and the y-intercept is (0, 2)
3 3
13) Slope is m= and the y-intercept is (0, -3) 4 −1
15) Slope is m= and the y-intercept is (0, 6) 4
17) 𝑦 = 𝑥 + 1 19) 𝑦 = 𝑥 + 2 21) 𝑦 = 𝑥 + 4 23) 𝑦 = 3𝑥 25) 𝑥 = 2 27) 𝑥 = −8 29) 𝑦 = 5 31) 𝑦 = 8 33) 𝑦 = 𝑥 + 2
35) 𝑦 = −4
𝑥 5
𝑥 8 37) 𝑦 =
3+
3
Section 11.1: Solving a System of 2 Equations in 2 Unknowns Graphically 1) (3, 1) 3) (1, 3)
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5) (2, 4) 7) (3, 1) 9) (1, 2) 11) (-3, 4) 13) (5, 0) 15) (6, 5) 17) Inconsistent 19) Dependent
Section 11.2: Solving a System of 2 Equations in 2 Unknowns by the Use of the Substitution Method
1) (4, 3) 3) (5, 2) 5) (1, 3) 7) (7, 2) 9) (4, 2) 11) (1, 3) 13) (3, -3) 15) (6, 7) 17) (2, 1) 19) (2, 2)
Section 11.3: Solving a System of 2 Equations in 2 Unknowns by the Use of the Elimination Method (Adding or Subtracting)
1) (2, 1) 3) (4, 3) 5) (3, -1) 7) (-5, 3) 9) (-2, 2) 11) (5, -3) 13) (-3, 4) 15) (6, 5) 17) (4, -3) 19) (-3, 4)
Section 12.1: Ratio and Proportion
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1) 4.8 gallons 3) 3 and 1/8 pounds 5) 16 ounces 7) a) $6; b) $9; c) $12 9) a) 90 students; b) 150 students; c) 200 students
Section 12.2: Percent Problems 1) 12 3) 6000 5) 27 students 7) 72 million 9) a) $160 reduction; b) $640 new price 11) a) $9 tip; b) $54 13) 4% 15) $18,200 17) a) $35 sales tax; b) $535 19) a) 800 students; b) 1200 students
Section 12.3: Number Problems 1) 18 3) 6 5) 1 7) 9 9) 10 11) 20 13) 13 15) a) 5 feet and 10 feet; b) 6 feet and 9 feet; c) 4 feet and 11 feet
Section 12.4: Distance = Rate X Time 1) 480 miles 3) 8 miles per hour 5) 1400 miles 7) 4 miles per hour 9) 385 miles
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11) a) 28.23 kilometers per day; b) 2.823 kilometers per hour 13) a) 29.25 kilometers per day; b) 2.925 kilometers per hour
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